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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 89232.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
89232.h1 | 89232bo2 | \([0, -1, 0, -19933944, 34221867120]\) | \(44308125149913793/61165323648\) | \(1209275738816836534272\) | \([2]\) | \(8128512\) | \(2.9497\) | |
89232.h2 | 89232bo1 | \([0, -1, 0, -897784, 840056944]\) | \(-4047806261953/13066420224\) | \(-258331093954499444736\) | \([2]\) | \(4064256\) | \(2.6031\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 89232.h have rank \(1\).
Complex multiplication
The elliptic curves in class 89232.h do not have complex multiplication.Modular form 89232.2.a.h
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.