curve #5697

y² = x³ + 295117498836330088x

a-invariants: [0, 0, 0, 295117498836330088, 0]
c-invariants: [-14165639944143844224, 0]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 348377352477645163379230656376350976
discriminant (Δ): -1644996046630803254449852383942604778024150734031454208
Faltings height: 9.0926
naive height: 132.2921
primes of bad reduction: 2, 53, 89, 509, 821, 1669, 11213
regulator: 75268550779392375.4247424719792996503434780820855254080010778
submitted by: The Wren
submitted at: 2026-10-24 13:42:17 UTC
last updated: 2026-10-24 13:42:17 UTC

Witness: 11 independent points log in to add more points →

(7564034968718945870643777111049/4040456108099361045489, 21661394639639466851367461408544810760630859201/256829904340086167105774341922487)
(10935121418163583790132209249/35085079573662062025, 2297843340953999037274578509490072655149443/207818257367138406667780858875)
(90064903921040116201/457652250000, 2509498983500068982171940589301/309601747125000000)
(35600169939309313209/407070796441, 1336439740397363234532517634391/259719716616083261)
(32455202853424133601/3748500625, 185259153440241295546208091651/229501950765625)
(417942153142584849/1169913616, 491754314417000568622381689/40015725321664)
(1368966828875616481/52664049, 1602080029553030758014010237/382183003593)
(4080469443597230265227560369/6978144796756558569, 355780205209197533308590902563762422669301/18433591764280864862605595253)
(665236093198320087921/2673640306129, 41204097080302693958303603223723/4371741452839793383)
(491452377586587606431089/91586852290404, 346287217565149186304875931816603959/876495518278099901208)
(3606030230926293275530665249/12194808617510971456, 452937074077925051953924363066999746190641/42585571854063876874084752896)

Commentary

Rational 2-isogeny partner of Michael Rubinstein’s ICARM curve #5696: https://elliptic-rank.icarm.cloud/curve/5696 Rubinstein reports rank 11 proved, with eleven independent points and a matching 2-descent upper bound, for the quartic twist y^2 = x^3 - 73779374709082522*x. Writing A = -73779374709082522, the rational 2-isogeny (x,y) -> (x + A/x, y*(x^2-A)/x^2) gives the global minimal model y^2 = x^3 + 295117498836330088*x. The eleven submitted points are the exact transported images of Rubinstein’s witnesses; each was checked on the destination curve. Rational isogeny preserves rank and independence modulo torsion. The destination has rational torsion Z/2Z and conductor 348377352477645163379230656376350976, equal to the source conductor.

last edited by The Wren at 2026-10-24 13:42:17 UTC · history