curve #698

y² + xy + y = x³ − x² − 390634775444173549902842x + 90963878019286462203814005481220665

a-invariants: [1, -1, 1, -390634775444173549902842, 90963878019286462203814005481220665]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 163775308591612518303223046086861799473733719535986646946
discriminant (Δ): 240431153704810409566340109843202606670306387398611482973714926134382592
Faltings height: 12.4664
naive height: 174.5798
primes of bad reduction: 2, 3, 7, 19, 37, 107, 23202396769, 107662818007, 6917356945540909971138072997
regulator: 39270749109476736.075744471935453083905570045793521210786979747794823755755
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:24 UTC
last updated: 2026-09-10 16:08:24 UTC

Witness: 18 independent points log in to add more points →

(412363327297, 362221000580135)
(17301431761, 290190473810558087)
(686058245521, 381937822971424871)
(722857562257, 431624844617525255)
(-719038742543, 9506586891306887)
(-434419027163, 422705251032229547)
(171777684433, 170088959540786087)
(203395778689, 141155362500258215)
(122401351441, 212093035694583047)
(225433599697, 119825925210739367)
(421988384753, 35577630466111367)
(506479875025, 151783762035119015)
(520038035473, 168694118791479047)
(-160525074173, 386696467811424845)
(-558070421519, 367639628577126791)
(71287575133, -251949930965747521)
(643303180273, 325408986344986247)
(7532813713, 296684553402688007)

Commentary

Found by a search of rational specializations of Mestre families, using finite-field scores to select fibers and rational-point searches to construct the witness. Point independence checked with the site verifier.

last edited by David Renshaw at 2026-09-10 16:08:24 UTC · history