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SageMath
E = EllipticCurve("hm1")
E.isogeny_class()
Elliptic curves in class 382200hm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
382200.hm2 | 382200hm1 | \([0, 1, 0, -408, -2061312]\) | \(-4/975\) | \(-1835324400000000\) | \([2]\) | \(1658880\) | \(1.6077\) | \(\Gamma_0(N)\)-optimal |
382200.hm1 | 382200hm2 | \([0, 1, 0, -245408, -46161312]\) | \(434163602/7605\) | \(28631060640000000\) | \([2]\) | \(3317760\) | \(1.9543\) |
Rank
sage: E.rank()
The elliptic curves in class 382200hm have rank \(1\).
Complex multiplication
The elliptic curves in class 382200hm do not have complex multiplication.Modular form 382200.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.