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SageMath
E = EllipticCurve("hm1")
E.isogeny_class()
Elliptic curves in class 462462.hm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
462462.hm1 | 462462hm3 | \([1, 1, 1, -11600212, 15199218761]\) | \(828279937799497/193444524\) | \(40318168107263682636\) | \([2]\) | \(23592960\) | \(2.7525\) | \(\Gamma_0(N)\)-optimal* |
462462.hm2 | 462462hm2 | \([1, 1, 1, -809432, 178453001]\) | \(281397674377/96589584\) | \(20131430989086392976\) | \([2, 2]\) | \(11796480\) | \(2.4060\) | \(\Gamma_0(N)\)-optimal* |
462462.hm3 | 462462hm1 | \([1, 1, 1, -335112, -72746871]\) | \(19968681097/628992\) | \(131096009696940288\) | \([2]\) | \(5898240\) | \(2.0594\) | \(\Gamma_0(N)\)-optimal* |
462462.hm4 | 462462hm4 | \([1, 1, 1, 2392228, 1243965449]\) | \(7264187703863/7406095788\) | \(-1543596111302077965132\) | \([2]\) | \(23592960\) | \(2.7525\) |
Rank
sage: E.rank()
The elliptic curves in class 462462.hm have rank \(1\).
Complex multiplication
The elliptic curves in class 462462.hm do not have complex multiplication.Modular form 462462.2.a.hm
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.