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SageMath
E = EllipticCurve("ii1")
E.isogeny_class()
Elliptic curves in class 463680.ii
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.ii1 | 463680ii2 | \([0, 0, 0, -243372, 25760464]\) | \(8341959848041/3327411150\) | \(635878173140582400\) | \([2]\) | \(5898240\) | \(2.1143\) | \(\Gamma_0(N)\)-optimal* |
463680.ii2 | 463680ii1 | \([0, 0, 0, -110892, -13930544]\) | \(789145184521/17996580\) | \(3439199995822080\) | \([2]\) | \(2949120\) | \(1.7678\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 463680.ii have rank \(1\).
Complex multiplication
The elliptic curves in class 463680.ii do not have complex multiplication.Modular form 463680.2.a.ii
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.