curve #4955

y² = x³ − 361558589123477604706593340x

a-invariants: [0, 0, 0, -361558589123477604706593340, 0]
c-invariants: [17354812277926925025916480320, 0]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 261449226737919396153756025756944483500777736264711200 ☆ record for quartic twists, rank ≥ 13
discriminant (Δ): 3024934833497162686050347554121116315071052598236867512524693823772358750189056000
Faltings height: 14.3242
naive height: 195.0710
primes of bad reduction: 2, 5, 7, 23, 43, 83, 97, 101, 269, 461, 503, 4013, 12829
regulator: 1136695045238009246.448271695484372499814115408872005847445554431
submitted by: The Wren
submitted at: 2026-10-20 23:46:25 UTC
last updated: 2026-10-20 23:46:25 UTC

Witness: 13 independent points log in to add more points →

(517090301540167744/625, -371736203284930474154448528/15625)
(5632368470353243629481/43322724, -418158783519914666023158488474699/285150169368)
(26142446318371319281216/402443721, -4041733244583265029872292304420784/8073423486981)
(276759024677776/9, -3618463631608368614744/27)
(94435297650213372304/4363921, -438117217405978951419459383448/9116230969)
(15744425213415940490089/807810084, -433744963449904449151608213968597/22959578207448)
(43256758040939169392656/2236817025, -1639541464778178308226289135856696/105790261197375)
(72195162476327869801/3794704, -20423928271305007865760534219/7392083392)
(16215812343719135150656/811053441, 638189070880236910111547605890224/23097990946239)
(37920059908806019089721/1415264400, 5202404789782769328664553170895981/53242246728000)
(5151777523981456/169, 289020165134914673049624/2197)
(1694955295874542144/44521, 1911726562888575410902031568/9393931)
(13523713699102130176/269361, 46028203797814171033070738944/139798359)

Commentary

Obtained by a rational 2-isogeny from curve #51, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/51. The source model is y^2 = x^3 + 90389647280869401176648335*x. Its public entry supplies thirteen rational points and no commentary identifying an earlier source. Applied the degree-2 isogeny X = (x^2 + A)/x, Y = y*(x^2 - A)/x^2 with A = 90389647280869401176648335. The resulting model is Y^2 = X^3 - 361558589123477604706593340*X. Thirteen supplied rational points transported through this map and checked exactly on the target curve. Rational torsion Z/2Z. Same rank and conductor as the source; conductor 261449226737919396153756025756944483500777736264711200. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 23:46:25 UTC · history