The elliptic curve rank, also called the Mordell-Weil rank, of an elliptic curve 
over 
is the group rank of its group
of rational points. By the Mordell-Weil
theorem,
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(1)
|
where 
is the finite subgroup of
torsion points and the nonnegative
integer 
is the rank (Silverman 1986). Equivalently, 
is the maximal number of linearly
independent rational points of infinite order.
It is not known whether the ranks of elliptic curves over 
are bounded. A traditional conjecture
says that they are unbounded. On the other hand, the heuristic
of Park et al. (2019) predicts that they are bounded
and that only finitely many elliptic
curves over 
have rank greater than 21.
As of Aug. 29, 2026, the largest known lower bound for the rank of an elliptic curve over 
is 31. The record elliptic curve,
submitted to the ICARM Elliptic Curve Rank Leaderboard on Aug. 23, 2026 and
attributed there to Claude, L. Alpöge, and A. Howell, is
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Thirty-one linearly independent rational points prove unconditionally that its rank is at least 31. The rank is reported to be exactly 31 assuming both the generalized Riemann hypothesis and the Swinnerton-Dyer conjecture (ICARM, curve 302).
Earlier records were rank at least 28, found by N. D. Elkies in 2006, and rank at least 29, found by Elkies and Z. Klagsbrun in 2024 (Dujella). On Aug. 20, 2026, the elliptic curve
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together with thirty linearly independent rational points, was submitted under the name "ranksunbounded" and subsequently attributed to Claude working with Alpöge and Howell (ICARM, curve 273).
Applying the analytic method of Bober (2013) gives an upper bound of 31 under the generalized
Riemann hypothesis for the multiplicity of 
as a zero
of the elliptic curve's Hasse-Weil
L-function. The sign in its functional
equation is 1, so this multiplicity is an even number. Consequently, assuming both the generalized
Riemann hypothesis and the Swinnerton-Dyer
conjecture, the rank of this elliptic curve
is exactly 30. Without these assumptions, only the lower
bound of 30 is known for this example.
Elkies and Klagsbrun independently found another elliptic curve of rank at least 30 in Sept. 2025, announced in Aug. 2026 (Elkies 2026; ICARM, curve 398),
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(6)
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(7)
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Its elliptic curve conductor 
and elliptic discriminant 
satisfy 
and 
, both smaller than those of the Aug. 20
example, for which 
and 
.
Elkies (2026) reports that its rank is exactly 30 under the generalized
Riemann hypothesis for number fields, without
assuming the Swinnerton-Dyer conjecture.
The thirty linearly independent rational
points give the unconditional lower bound of 30.
The same announcement gives further examples with small elliptic curve conductors or small elliptic discriminants
and reported ranks 27, 28, and 29. In particular, the example of rank at least 28
cataloged as ICARM curve 400 has 
and 
. Updated records and their rational
points are maintained by ICARM.
The largest rank of an elliptic curve whose exact rank is known unconditionally is 20 (Elkies and Klagsbrun 2020, Dujella).
and 
Records for Elliptic Curves 
of Ranks 27-30; Record 2-Ranks 25 (Real) and 26 (Complex)
for Class Groups of Cubic Fields." Number Theory List, Aug. 29, 2026. 
Pegg, E. Jr. "New Record in Elliptic Curves by Elkies-Klagsbrun:
Rank 29 (and Now 30)." Wolfram Community, Aug. 20, 2026.