curve #4010

y² + xy + y = x³ − x² + 329486755683181567632331001848x + 465867166506616063474137421171883715144891979

a-invariants: [1, -1, 1, 329486755683181567632331001848, 465867166506616063474137421171883715144891979]
c-invariants: [-15815364272792715246351888088719, -402509231861716350010793959459726138468683069321]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 12334433622321215895724716304610404854359954238330
discriminant (Δ): -96047171028935992490827439854997921398381870379181168715638087486705253134164416716897600000
Faltings height: 16.3300
naive height: 219.2281
primes of bad reduction: 2, 3, 5, 7, 11, 37, 43, 59, 67, 97, 137, 151, 157, 449, 631, 11551, 172351, 356533, 66670921
regulator: 48314554385602796533121.138636269991148501595676012142499075497385233
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:45:05 UTC
last updated: 2026-10-05 04:45:05 UTC

Witness: 15 independent points log in to add more points →

(-188628315440386890943/516961, 6406380700280005344189534038179/371694959)
(-42326073377526933219601/136212241, 29030805033529227510729559774501057/1589733064711)
(406869418669332689/961, 777572211194692969798802543/29791)
(11063220912160885303/15129, 61657378202238319942591964027/1860867)
(-5648013310341024527028510511842707/9582699073903941049, 242664361237581238949477851387710687565455148149877/29664140898554166018339994157)
(497442497498485852796194312207/3287018537540676, 4294068418055031077128910510057780093533978181/188453075766032782827576)
(-42784429875828653175156127/81588495769, 284360157140235711998779860245827325953/23304693165969853)
(310784355239063069573/1054729, 26278341152227399078516362164723/1083206683)
(489707948007750809522686213/508550118129, 14847235079877344768756673061602090982543/362660820090979383)
(621801173476972287817891487/14023519550401, 1151227185908140459161179617381345845952969/52515261988822114399)
(908172042911543833368498065453/369517154763241, 901888043680029518350425534800345484173563397/7103165078580317228789)
(11510368974701803752079736926927/5146575803850321, 41096863252812961492128609602662354182633209089/369213490254356838574119)
(7453009332098457113231879119389721/45875148684413409, 643428407649376215964043791771941464645470390208111/9825761702073337291413423)
(436766820470065275597154657591/11848534308900, 288688260266715264307996780632269357828013179/40784669342066313000)
(-49438395590173085781600097777/94218025320409, 11144717712154436770323649635922617568883683/914536401921006038173)

Commentary

Unlisted rational-isogeny neighbor of curve#3645 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 15: 15 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:45:05 UTC · history