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curve #1209

y2 + xy + y = x3 − 384893316356139442x + 91567955626994363782396164
a-invariants
[1, 0, 1, -384893316356139442, 91567955626994363782396164]
rank (lower bound)
≥ 12
torsion subgroup
ℤ/2ℤ
conductor (N)
13361790818375694829477764814281226
discriminant (Δ)
27042412530937835487705478809524704801262020180377152 ☆ record for ℤ/2ℤ torsion, rank ≥ 12
Faltings height
8.9145
naive height
133.0888
primes of bad reduction
2, 13, 29, 41, 71, 73, 107, 149, 293, 857, 1193, 180241, 96875897
regulator
1281489280176.3484702951801664751442176085584953152223415628141
submitted by
David Renshaw
submitted at
last updated

Witness: 12 independent points log in to add more points →

Commentary

Julian Aguirre, Alvaro Lozano-Robledo, and Juan Carlos Peral, Elliptic curves of maximal rank, Example 1.4(4) and Corollary 3.6: https://production.wordpress.uconn.edu/alozano/wp-content/uploads/sites/490/2014/01/ALP-2-23-07.pdf . Their model is y^2 = x^3 + 4510328029*x^2 + 622726581362777216*x. Submitted in its global minimal model, with 12 rational points computed independently using PARI/GP ellrank. PARI returns rank bounds [12,12] and torsion invariants [2], proving rank exactly 12 and torsion Z/2Z. The supplied points are checked by the ICARM exact independence verifier. Found through Lozano-Robledo's own blog: https://afieldguidetomath.wordpress.com/2020/04/29/birding-and-math/ .

last edited by David Renshaw at · history

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