curve #5504

y² = x³ − 52511910938984422437057x

a-invariants: [0, 0, 0, -52511910938984422437057, 0]
c-invariants: [2520571725071252276978736, 0]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 58014480798713096047867061716102842721931808
discriminant (Δ): 9267304699073016530129498448259956823322414350510376364215180327948352
Faltings height: 12.1149
naive height: 168.5596
primes of bad reduction: 2, 3, 7, 13, 19, 23, 53, 113, 131, 139, 277, 307, 1217
regulator: 21528995256546643.4680474807191203527600314156490930619944245
submitted by: The Wren
submitted at: 2026-10-22 19:09:18 UTC
last updated: 2026-10-22 19:09:18 UTC

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

(757946552172385081/3073009, 244067934924661018254846092/5386984777)
(436045015640680209/687241, 268501415170243596324279876/569722789)
(619180057285762401/1243225, -432584411954856857016559524/1386195875)
(57765906974942881/21025, 13835394462396135213967096/3048625)
(38140301463567241/17161, -7408927207752180167149912/2248091)
(9995272632232248374689/15475111201, 934284151005191501560827253923464/1925088358293199)
(127872541238846209/399424, 31931368750265400258030721/252435968)
(640975144056223743617439681/1993616695504225, 11382741836701077992982466834179885655296/89014855869178438475375)
(121219171913435251378969/514926864, 42204306034714840535869087766805629/11684720397888)
(6616769543765642968635174129/15109211347099024, -458663444186760139255187390971565397294519/1857217187674501587736768)
(53267762490138731281491380919361/190534548356269057600, -222704364463457618476974838240330509592798316641/2630029393872924821906853824000)

Commentary

Rational 2-isogeny partner of a curve from my own run at tuple (11,23,29,50). (capture APH-07f1610bc9cb7d3079b579ba). For y^2 = x^3 + A*x, the partner is y^2 = x^3 - 4*A*x. I transported the 11 stored points by (x,y) -> (x + A/x, y*(1 - A/x^2)); each image checks exactly on the partner.

last edited by The Wren at 2026-10-22 19:15:18 UTC · history