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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 485100.ec
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485100.ec1 | 485100ec1 | \([0, 0, 0, -334425, -75331375]\) | \(-3937024/55\) | \(-57784924023750000\) | \([]\) | \(5225472\) | \(2.0223\) | \(\Gamma_0(N)\)-optimal* |
485100.ec2 | 485100ec2 | \([0, 0, 0, 1209075, -377857375]\) | \(186050816/166375\) | \(-174799395171843750000\) | \([]\) | \(15676416\) | \(2.5716\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 485100.ec have rank \(0\).
Complex multiplication
The elliptic curves in class 485100.ec do not have complex multiplication.Modular form 485100.2.a.ec
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.