Elliptic Curve Rank Leaderboard

curve #0

y2 + y = x3 − x2
a-invariants
[0, -1, 1, 0, 0]
rank (lower bound)
≥ 0
torsion subgroup
ℤ/5ℤ
conductor (N)
11 ★ record for rank ≥ 0
discriminant (Δ)
-11 ★ record for rank ≥ 0
Faltings height
-1.1127 ★ record for rank ≥ 0
naive height
10.0478 ★ record for rank ≥ 0
primes of bad reduction
11
regulator
1
submitted by
David Renshaw
submitted at
last updated

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Commentary

Cremona 11a3 = LMFDB 11.a3: y² + y = x³ − x², the modular curve X₁(11). https://www.lmfdb.org/EllipticCurve/Q/11/a/3 The smallest elliptic curve over ℚ by every measure this board tracks: conductor 11 (the smallest possible, shared with its isogeny class), minimal discriminant −11 (|Δ| = 11 is the smallest possible, shared with 11a2), and the strictly smallest naive height (log 152² ≈ 10.048) and Faltings height (≈ −1.1127). Its Mordell–Weil rank is exactly 0 and its torsion subgroup is ℤ/5ℤ, generated by (0, 0), so no witness can ever raise its rank bound. Seeded as curve #0, the board's rank-0 anchor.

last edited by David Renshaw at · history

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