Largest known rank of an elliptic curve
Relevant papers:
* The rank of elliptic curves by Armand Brumer, Kenneth Kramer
* Ranks of elliptic curves by Karl Rubin, Alice Silverberg
* Three lectures on elliptic surfaces and curves of high rank by Noam D. Elkies
* New rank records for elliptic curves having rational torsion by Noam D. Elkies, Zev Klagsbrun
The question of the largest possible rank of an elliptic curve over the field of rational numbers, \(\mathbb{Q}\), is a major open problem in number theory. By the Mordell-Weil theorem, the group of rational points on an elliptic curve \(E(\mathbb{Q})\) is a finitely generated abelian group, isomorphic to \(T \oplus \mathbb{Z}^r\), where \(T\) is the finite torsion subgroup and \(r\) is the rank. It is conjectured that the rank can be arbitrarily large, but this remains unproven. The search for elliptic curves with high rank is an active area of research that periodically yields new records.
In August 2024, Noam D. Elkies and Zev Klagsbrun announced the discovery of an elliptic curve with a rank of at least 29. This broke the long-standing record for a curve with rank at least 28, found by Elkies in 2006.
The equation for the rank \(\ge 29\) curve is:
\(y^2 + xy = x^3 - 27006183241630922218434652145297453784768054621836357954737385x + 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497\)
The previous record-holder, with a rank of at least 28, was found by Noam D. Elkies in 2006 and has the equation:
\(y^2 + xy + y = x^3 - x^2 - 20067762415575526585033208209338542750930230312178956502x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429\)
These ranks are established by exhibiting a sufficient number of independent rational points, which provides a lower bound. Proving the exact rank is computationally very difficult. However, assuming the Generalized Riemann Hypothesis (GRH), the rank of the 2006 curve is exactly 28. Similarly, the new curve’s rank is believed to be exactly 29, conditional on GRH. The method used to find these record-breaking curves involves constructing and analyzing specialized elliptic K3 surfaces.
Separate from the overall record, there is significant interest in finding high-rank curves with specific, non-trivial torsion subgroups. By Mazur’s Theorem, there are only 15 possible torsion subgroups for an elliptic curve over \(\mathbb{Q}\). Some notable records for curves with prescribed torsion include:
The largest known rank of an elliptic curve over the rational numbers (\(\mathbb{Q}\)) is at least 29. This record was set in August 2024 by Noam Elkies and Zev Klagsbrun, who constructed an elliptic curve and exhibited 29 independent rational points of infinite order.
This discovery surpassed the previous record, which had stood for 18 years. In 2006, Noam Elkies found an elliptic curve with rank at least 28. The rank of this curve was later proven to be exactly 28, conditional on the truth of the Generalized Riemann Hypothesis (GRH), in a paper by Klagsbrun, Sherman, and Weigandt. The highest rank for an elliptic curve over \(\mathbb{Q}\) that has been proven exactly and unconditionally is 20.
The rank of an elliptic curve refers to the number of independent generators of its group of rational points, known as the Mordell-Weil group, \(E(\mathbb{Q})\). This group is isomorphic to \(T \oplus \mathbb{Z}^r\), where \(T\) is the finite torsion subgroup and \(r\) is the rank. Finding curves with high rank is a significant computational challenge. It is a major open problem in number theory whether the rank of elliptic curves over \(\mathbb{Q}\) is bounded or can be arbitrarily large.
The largest known rank of an elliptic curve over \(\mathbb{Q}\) is at least 28. This is based on a construction by Noam Elkies in 2006, who found an elliptic curve with rank at least 28. The exact rank of this curve is not known, but it is at least 28.
Here is the curve with rank \(\geq 28\):
The coefficients are quite large, and the full equation is:
This curve is constructed to have many independent rational points, which implies a high rank. However, computing the exact rank of an elliptic curve is a difficult problem, and in this case, it is only known that the rank is at least 28.
It is not known whether elliptic curves over \(\mathbb{Q}\) can have arbitrarily large rank, but it is conjectured that they can. The current record is the curve with rank \(\geq 28\). There are also examples of elliptic curves over number fields (other than \(\mathbb{Q}\)) with much higher ranks. For example, curves with rank at least 40 have been found over certain number fields.
The rank of an elliptic curve is a fundamental invariant in number theory, related to the number of independent rational points of infinite order on the curve. It is a measure of the size of the group of rational points on the curve, as described by the Mordell-Weil theorem, which states that the group of rational points on an elliptic curve over the rationals (ℚ) is finitely generated and isomorphic to ℤ^r ⊕ T, where r is the rank and T is a finite torsion subgroup.
As of the latest research (up to October 2023), the largest known rank of an elliptic curve over the rationals is 28. This record was achieved by Noam D. Elkies in 2006, who constructed an elliptic curve with rank at least 28. The curve was found using sophisticated computational techniques and number-theoretic methods, often involving searching for curves with many rational points or using modular forms and L-functions to predict high-rank curves.
Elkies’ curve with rank 28 is defined by a specific Weierstrass equation, and its rank was verified computationally by finding 28 independent rational points of infinite order. The minimal model of such a curve typically has very large coefficients, reflecting the complexity of high-rank curves.
The largest known rank of an elliptic curve over the rationals is 28, due to Noam D. Elkies in 2006. This is likely not the ultimate limit, and ongoing research in computational number theory and elliptic curve theory may yield curves with even higher ranks in the future. If you are interested in the specific equation of Elkies’ curve or computational methods to find high-rank curves, I can provide further details or references to relevant papers and resources.
The largest known rank of an elliptic curve over the rational numbers is at least 29.
This record is currently held by an elliptic curve discovered by Noam Elkies and Zev Klagsbrun in 2024. They found an elliptic curve over ℚ with at least 29 independent rational points.
Some key points about elliptic curve ranks:
Previous records: Before this, the record was 28 (found in 2006 by Elkies), and before that 24 (found by Martin-McMillen in 2000)
The rank conjecture: It’s unknown whether elliptic curve ranks over ℚ are bounded or unbounded. The Birch and Swinnerton-Dyer conjecture relates to this question
Computational challenges: Finding high-rank curves is computationally intensive, requiring sophisticated search strategies and substantial computing power
Typical ranks: Most elliptic curves have rank 0 or 1, and ranks above 3 are quite rare
The search for high-rank elliptic curves remains an active area of research in arithmetic geometry, with improvements in both theoretical understanding and computational methods continuing to push the boundaries of what we can discover.
The largest known rank of an elliptic curve over \(\mathbb{Q}\) is at least 28, as demonstrated by a curve discovered by Noam Elkies in 2006.
The curve is defined by a complex equation, but its construction involves sophisticated techniques (e.g., searching in families with high rank potential).
It remains unknown whether elliptic curves over \(\mathbb{Q}\) can have arbitrarily large rank, though this is widely believed.
For further reading, see:
- Elkies’ work on high-rank curves.
- The Brumer-McGuinness heuristic for rank distribution.
Would you like details on the explicit equation or construction methods?
The largest known rank of an elliptic curve defined over the rational numbers is 28. This record is held by a curve discovered by Noam Elkies in 2006. The equation of this elliptic curve is:
This record stands as of 2023, reflecting ongoing research in elliptic curve theory. For further exploration, databases like the LMFDB (L-functions and Modular Forms Database) catalog high-rank curves.
As of my knowledge cutoff in October 2023, the largest known rank of an elliptic curve over the rational numbers ℚ is 28. This remarkable achievement was accomplished by Noam Elkies in 2006.
An elliptic curve \( E \) defined over the rationals \( \mathbb{Q} \) can be represented by an equation of the form:
where \( a, b \in \mathbb{Q} \). According to the Mordell-Weil Theorem, the set of rational points \( E(\mathbb{Q}) \) forms a finitely generated abelian group. This group can be expressed as:
where:
- \( T \) is a finite torsion subgroup.
- \( \mathbb{Z}^r \) represents the free part, and \( r \) is the rank of the elliptic curve.
The rank \( r \) essentially counts the number of independent infinite-order rational points on the curve. Determining the rank of an elliptic curve is a central problem in number theory and has deep connections to various areas, including the Birch and Swinnerton-Dyer conjecture.
Noam Elkies (2006): Elkies constructed an elliptic curve over \( \mathbb{Q} \) with rank 28, setting the current record. His construction involved intricate methods combining both theoretical insights and computational techniques.
Subsequent Efforts: Since Elkies’ breakthrough, several mathematicians have worked on finding elliptic curves with higher ranks. While there have been claims and constructions suggesting higher ranks, many either have not been fully verified or involve curves over number fields rather than strictly over \( \mathbb{Q} \).
Unboundedness Conjecture: It is conjectured that the rank of elliptic curves over \( \mathbb{Q} \) is unbounded, meaning that for any positive integer \( N \), there exists an elliptic curve over \( \mathbb{Q} \) with rank at least \( N \). However, this remains unproven.
Average Rank: Research by Bhargava and Shankar has shown that the average rank of elliptic curves over \( \mathbb{Q} \) is likely bounded, specifically suggesting an average rank around 1/2.
High-rank elliptic curves are of significant interest not only for theoretical reasons but also for their applications in areas like cryptography and integer factorization. However, constructing such curves typically requires advanced computational tools and deep theoretical insights.
The quest to determine the largest possible rank of elliptic curves over \( \mathbb{Q} \) continues to be a vibrant area of research in number theory. As computational techniques and theoretical methods advance, it’s plausible that higher ranks may be discovered in the future, potentially reshaping our understanding of these fascinating mathematical objects.
Okay, so I need to figure out what the largest known rank of an elliptic curve is. Hmm, elliptic curves are a big topic in number theory, right? I remember they’re equations of the form y² = x³ + ax + b, with some conditions on a and b to ensure the curve is non-singular. The rank part is about the group of rational points on the curve. The rank tells you how many independent rational points of infinite order there are. So, higher rank means more such points. But what’s the highest rank that’s been found so far?
I think I’ve heard that there are elliptic curves with rank as high as 28, but I’m not sure if that’s still the record. Maybe it’s even higher? Or maybe it’s lower? Wait, I should check some sources or recall some famous mathematicians who worked on this. I remember that Noam Elkies is someone who has done a lot of work with elliptic curves and high-rank examples.
Let me think. The Birch and Swinnerton-Dyer conjecture relates the rank to the order of vanishing of the L-function at s=1. But that’s more theoretical. For explicit examples, constructing elliptic curves with high ranks is challenging. I think there were some families of elliptic curves where people managed to find high ranks through clever constructions. Maybe using K3 surfaces or something like that?
Wait, there was an example by Martin and McMillen in 2000 who found an elliptic curve with rank at least 24, right? Then later, in 2006 or so, Noam Elkies found one with rank 28? Or was that the current record? Or maybe there’s been more recent progress?
I should also recall that the exact rank is hard to compute, so sometimes people can only determine a lower bound for the rank. So maybe the highest known rank is 28, but it might have an even higher rank which hasn’t been proven yet. Or maybe there are curves with higher ranks but they haven’t been fully verified?
Another thing to consider is whether we’re talking about over the rational numbers or over other fields. The question says “elliptic curve,” which without qualification usually refers to over the rational numbers, Q. But sometimes people study elliptic curves over function fields or other number fields, where the ranks can be higher. But the problem here is probably asking about Q.
So, going back. In 2000, the record was maybe 24. Then Elkies in 2006 found a curve with rank at least 28. Let me check some references. Wait, in 2006, Elkies discovered an elliptic curve with rank at least 28. But is that still the record? Or has someone found a higher rank since then?
I think I’ve heard that the highest known rank was 28, but maybe it’s 19? No, that seems low. Wait, confusion arises because sometimes the rank is conditional on the Birch and Swinnerton-Dyer conjecture or other assumptions. If the rank is computed unconditionally, maybe the number is different.
Wait, let me recall. The rank of an elliptic curve over Q is the number of independent generators of infinite order in its Mordell-Weil group. To compute the rank, you can sometimes use the method of descent, which gives an upper bound on the rank, and then you find explicit points to get a lower bound. If the bounds coincide, you get the exact rank.
So, in the case of high-rank curves, they might have a lower bound of rank 28, but the upper bound might be hard to compute, so maybe they can’t prove it’s exactly 28. But if they found 28 independent points, then the rank is at least 28.
I think in Elkies’ case, he used a K3 surface to find elliptic curves with high ranks. There was a construction that allowed him to find many rational points. Alternatively, there’s a family of elliptic curves with high ranks. Also, there’s a result that says that the rank can be unbounded in certain families, but over Q, it’s not known whether the ranks can be arbitrarily large.
Wait, but the problem is about the largest known rank. So even if we don’t know if it’s bounded or not, what’s the highest one that’s been found so far.
Looking up in my mind… I think in 2023, the highest known rank is still 28, found by Elkies. But wait, another name comes to mind: Mordell. But that’s historical. Or maybe someone else?
Wait, actually, in 2023, I think the record is still 28. Wait, but I’m not sure. Maybe I should check some papers or online resources. Wait, but since I can’t actually access external resources, I need to rely on my memory.
Wait, another thought. There was a paper by Dujella in 2006 or 2007 where he surveys high-rank elliptic curves, and he lists the curve with rank 28 as the highest known. Also, in some other sources, it’s mentioned that the rank record hasn’t been surpassed for almost 20 years. But maybe there was a recent breakthrough?
Alternatively, maybe someone has found a curve with higher rank. Wait, I remember that in 2020 or so, there was a paper by some researchers claiming a higher rank, but I’m not sure. Alternatively, maybe the rank 28 curve is still the record.
Wait, here’s another angle. The question says “Largest known rank of an elliptic curve”. The key word is “known”. There’s a difference between the rank being “at least r” and “exactly r”. If they can’t compute the upper bound, then they can only say it’s at least r. So in the case of Elkies’ curve, he has 28 independent points, so rank at least 28. But the upper bound via descent might be much higher or not computable. However, in practice, if you have a lower bound of 28, and you conjecture that’s the exact rank, but unless you can prove it, then strictly speaking, the rank is at least 28.
But maybe in some cases, they can do a more detailed analysis. Wait, but in the literature, sometimes they list lower bounds for ranks. So if the question is about the largest known rank, it might be the highest lower bound achieved, so 28.
Alternatively, maybe someone has found a curve with a higher lower bound. Let me recall. I think in 2023, the highest known rank is still 28. Wait, but I should also remember that in 2022, there was a paper where they found a different curve with higher rank? Hmm.
Wait, here’s another thought. In 2006, Elkies found a curve with rank at least 28. Then in 2021, a paper by some authors (I can’t recall names) claimed a rank of 30, but maybe that’s not verified yet. Or maybe that’s in a different setting.
Alternatively, perhaps over a different field. Wait, but the question didn’t specify the field. But usually, in math problems like this, unless specified, it refers to over Q.
Wait, here’s a way to check. The famous mathematician Karl Rubin mentioned in some lecture that the rank record was 28. But that was maybe in 2015.
Alternatively, I can think of the following: the average rank of elliptic curves is conjectured to be 1/2, and most curves have low rank. However, there are some special constructions with high ranks.
Wait, here’s another data point. In 2016, Dujella and others maintained a list of high-rank elliptic curves. According to that list, the highest known rank is 28. So unless there’s been a more recent update, that might still be the case.
Wait, but if I can’t confirm a higher number, then maybe 28 is still the answer.
Wait, but I want to be careful here. Let me recall that in the 1990s, the record was 14 by Neron. Then in the early 2000s, it was 24, then Elkies pushed it to 28. Since then, no one has found a higher one?
Alternatively, there’s a curve called the “Dujella’s curve” which has rank 17 or something, but that’s not the record.
Alternatively, in 2018, there was a preprint where someone used a new method to construct high-rank curves, but I don’t know the details. But unless it’s widely accepted, maybe the record is still 28.
Alternatively, think about the method. Elkies used a K3 surface to find a curve with high rank. The idea is that sections on the K3 surface correspond to points on the elliptic curve. By finding many sections, you can get many independent points.
Alternatively, there’s a construction using Diophantine equations to create curves with lots of independent points.
But unless there’s a newer construction, 28 might still be the record.
Wait, but here’s another thought. In 2019, a paper by Park, Poonen, Voight, and Wood gave a heuristic suggesting that the ranks of elliptic curves over Q are bounded, with the upper bound possibly around 21 or something. If that’s the case, then maybe the rank
The largest known rank of an elliptic curve is a topic of ongoing research in number theory.
As of my cut-off knowledge in 2023, the largest known rank of an elliptic curve is at least 28. This was achieved by Elkies and Klagsbrun in 2020 using a combination of theoretical and computational techniques.
To be more specific, they found an elliptic curve over the rational numbers with rank at least 28. The curve is given by:
y^2 + xy + y = x^3 - x^2 - 20067762415575526585033208209338542750930230312178956502x + 344816117950305564670329856903907203748459679093647583812340895134514015292167*
This curve has rank at least 28, as verified by Klagsbrun using a rigorous computation.
It’s worth noting that the problem of determining the rank of an elliptic curve is a difficult one, and there is no known algorithm that can efficiently compute the rank for all curves. The Birch and Swinnerton-Dyer Conjecture, one of the seven Millennium Prize Problems, provides a framework for understanding the rank of elliptic curves, but it remains unsolved.
While we know that there are elliptic curves with rank at least 28, it’s still an open question whether there are curves with even higher rank. Researchers continue to explore this problem using a combination of theoretical and computational methods.
References:
Keep in mind that the state of knowledge on this topic may have evolved since my cut-off date. For the most up-to-date information, I recommend checking the latest research publications and online resources.
The largest known rank of an elliptic curve over the rational numbers (\(\mathbb{Q}\)) is currently 28.
This curve was discovered by Noam D. Elkies in 2000. Finding elliptic curves with high rank is an extremely difficult computational and theoretical challenge. The specific equation of such a high-rank curve often involves very large integer coefficients.
While the existence of curves with arbitrarily high rank is conjectured (e.g., by the Goldfeld-Hoffstein conjecture in certain forms), finding them becomes exponentially harder as the rank increases. This record stands as a significant achievement in computational number theory.
It’s important to note that this doesn’t contradict the Birch and Swinnerton-Dyer (BSD) conjecture, which relates the rank of an elliptic curve to the behavior of its L-function at \(s=1\). The BSD conjecture postulates that the analytic rank (derived from the L-function) is equal to the algebraic rank (the Mordell-Weil rank).
The largest known rank of an elliptic curve over the rational numbers, as of 2023, is 30. This record was achieved by Noam Elkies and Daniel J. Bernstein in 2021. The curve they discovered has a Mordell-Weil rank of 30, meaning its group of rational points is generated by 30 independent points.
This result is a milestone in number theory, contributing to our understanding of elliptic curves and their applications in cryptography and arithmetic geometry. For the most up-to-date information, consult databases like the LMFDB (L-Functions and Modular Forms Database).
Answer: The largest known rank of an elliptic curve over \(\mathbb{Q}\) is 30.
Over Q (the field in which most tables and “records” are kept)
• Proven Mordell–Weil rank record: 28
– Found by Noam D. Elkies (2006, communicated 2007).
– One model of his curve is
E : y² + xy + y = x³ − x² − 200 67762415575526585033208209338542771244320 x
+ 344 81631998931117885022575179694130985408463667501220508176640
Together with 28 independent rational points explicitly exhibited, the rank is unconditional.
What about “larger” ranks?
• Quadratic and higher number fields
– The present record over a quadratic field is 30 (Mestre, 1992; Q(√5)).
– Over higher–degree number fields still larger ranks are known (e.g. rank 34 over a degree-6 field, Elkies-Mestre).
• Function fields (e.g. Q(t))
– Ranks are unbounded; explicit constructions with rank ≥ 70 are known (Shioda, Ulmer, etc.).
• Analytic/“BSD-conditional” records
– Curves whose analytic rank is ≥ 34 are known, but the rank is only conditional on BSD, so they are not counted as official records.
Bottom line
As of today the largest proven Mordell–Weil rank for an elliptic curve over the rational numbers is 28.
For elliptic curves over Q the largest proven Mordell–Weil rank at present is 28. In 2006 Noam Elkies exhibited an explicit curve E/Q with
rank E(Q) ≥ 28. No unconditional example of higher rank is known, and it remains open whether ranks of elliptic curves over Q are unbounded.
Here are a few more details and references:
Record curve (Elkies, 2006)
One convenient model is given by the Weierstraß equation
y² + x y + y = x³ – 42 051 and some enormous coefficients…
(The full set of coefficients has hundreds of digits; see Elkies’s preprint below.)
History of the record over Q
• 1974 – Mestre found rank ≥ 11.
• 1989 – Stewart–Top, rank ≥ 12.
• 1992 – Kojima, rank ≥ 13.
• 1996 – Mestre, rank ≥ 14.
• 2000 – Elkies, rank ≥ 15.
• 2004 – Elkies, rank ≥ 17.
• 2005 – Elkies, rank ≥ 19.
• 2006 – Elkies, rank ≥ 28 (current record).
Why we don’t know a bound
There is no theorem limiting the rank of E(Q) from above. Conjecturally (Goldfeld, Katz–Sarnak heuristics) ranks should be “typically” 0 or 1 and unbounded families are scarce, but no proof of unboundedness or of any absolute bound is known.
Related records over function fields
Over Q(t) (the rational function field), one can exhibit nonconstant families with “generic” rank at least 18 (Dujella, 2014). Over Fq(t) yet larger ranks have been achieved.
For the explicit rank-28 example and its 28 independent rational points see
N. D. Elkies, “The existence of infinitely many elliptic curves of rank ≥ 28,” preprint, 2006, 28 pp., arXiv:math/0605162.
Excellent question. This is a famous and active area of research in number theory.
As of today, the largest known rank for a specific elliptic curve over the field of rational numbers (\(\mathbb{Q}\)) is at least 28.
The curve is given by the following equation:
\(y^2 + xy + y = x^3 - x^2 - 2006776241557552658503320820933854275093023239361064475310590479637881x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429\)
It’s important to note the phrase “at least 28”. Elkies demonstrated the existence of 28 independent rational points of infinite order on this curve. Proving that the rank is exactly 28 is a much harder problem and is currently computationally infeasible.
To understand why this is a significant record, it’s helpful to know what “rank” means.
The Mordell-Weil Theorem states that the set of all rational points on an elliptic curve \(E\), denoted \(E(\mathbb{Q})\), forms a finitely generated abelian group. This means the group structure looks like:
\(E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus T\)
A rank of 0 means there are only a finite number of rational points. A positive rank means there are infinitely many. A higher rank implies a much richer and more complex arithmetic structure.
Finding high-rank curves is not a matter of random searching. The methods are highly sophisticated and constructive. Elkies’s method involved:
The search for high-rank curves has a long history, with the record being broken incrementally over decades.
Rank ≥ | Discoverer(s) | Year |
---|---|---|
… | ||
12 | Mestre | 1992 |
14 | Mestre | 1992 |
15 | Fermigier | 1994 |
17 | Martin & McMillen | 1997 |
19 | Martin & McMillen | 1997 |
20 | Fermigier | 1998 |
21 | Dujella | 1999 |
22 | Martin & McMillen | 2000 |
23 | Martin & McMillen | 2000 |
24 | Martin & McMillen | 2000 |
28 | Elkies | 2006 |
This shows how significant Elkies’s jump from 24 to 28 was, breaking a long-standing record by a large margin.
The largest known rank of an elliptic curve over the rational numbers is 20.
This was found by Alexander Dujella in 2006. Specifically, he found an elliptic curve with the following equation:
y^2 + xy + y = x^3 - x^2 - 3136x - 57037
The rank of this curve is 20.
The rank of an elliptic curve is a fundamental property that is related to the number of points on the curve. It is also connected to other areas of number theory, such as the Birch and Swinnerton-Dyer Conjecture.
Finding curves with large rank is an active area of research, and it is not known if there exist elliptic curves of arbitrarily large rank over the rational numbers.
Rank 20 is currently the largest known rank for an elliptic curve over the rational numbers. However, new discoveries may change this in the future.
Would you like to know more about elliptic curves or number theory?
The largest known rank of an elliptic curve over the rational numbers Q is currently 28.
Rank 28: The curve with the highest known rank was found by Noam Elkies in 2006:
y² = x³ - x² - 20067762415575526585033208209338542750930230312178956502x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429
This curve has rank exactly 28, meaning its group of rational points has a free abelian subgroup of rank 28.
The search for high-rank elliptic curves has seen steady progress:
- Rank 24: Found by Elkies in the 1990s
- Rank 26: Found by Dujella and others in the early 2000s
- Rank 27: Found by Elkies around 2005
- Rank 28: Current record by Elkies (2006)
This relates to several deep conjectures:
Finding and verifying high-rank curves is computationally intensive because:
- You must prove the rank is exactly the claimed value (not just a lower bound)
- The coefficients of high-rank curves tend to be very large
- Verification requires computing the Mordell-Weil group structure
The question of whether elliptic curve ranks over Q are bounded remains one of the major open problems in arithmetic geometry.
The rank of an elliptic curve over the rational numbers (ℚ) is a fundamental invariant in number theory. It represents the rank of the Mordell-Weil group, which is the group of rational points on the curve minus the torsion subgroup. In simpler terms, it measures the number of “independent” rational points of infinite order on the curve. Higher ranks are of great interest because they provide insights into deep questions in mathematics, such as the Birch and Swinnerton-Dyer Conjecture.
As of my last knowledge update in 2023, the largest known rank for an elliptic curve over ℚ is 28. This record is held by certain elliptic curves, such as the one with Cremona label 148848h1, which was identified through computations by mathematicians like Noam Elkies and others. These results are often documented in databases and research papers.
However, mathematical research in this area is ongoing and highly active, so it’s possible that higher ranks have been discovered or verified since then. For the most current information, I recommend checking the following resources:
If you’re interested in learning more about elliptic curves or how ranks are computed, I can provide additional explanations or resources!
Over the rational numbers ℚ the best‐known explicit example is an elliptic curve with (proven) rank 28. This record–which was constructed by Noam Elkies (among others contributing to the high–rank search)–has stood for a while as the largest rank known for an elliptic curve over ℚ.
A few additional remarks to put this in perspective:
• It is widely believed (and supported by various heuristics) that there is no uniform bound on the possible rank of elliptic curves over ℚ, so one might expect there are curves with arbitrarily high rank. However, no one has been able to prove, or even construct, infinitely many examples with ever–increasing ranks.
• Much of the work on high–rank elliptic curves involves extensive computer searches and highly non–trivial constructions (often using families of curves with special properties). The record of 28 reflects the current state of these computations and constructions.
• The folklore in the community (and several survey articles on the topic) makes it clear that while many believe the rank is unbounded, the known examples with high rank are scarce and tend to become very complicated as the rank increases.
So, to answer your question directly: the largest known rank of an elliptic curve over ℚ is 28.
As of current knowledge, the highest known (proven) rank of an elliptic curve over the rationals is 28. In particular, Noam Elkies found an explicit example of a curve of rank 28 in the 1990s, and despite extensive searches since then, no one has produced a provably higher rank. It remains an open problem in number theory both whether there is a finite upper bound on the rank of elliptic curves over ℚ, and, if not, how large individual ranks can get in practice.