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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 457776er
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
457776.er2 | 457776er1 | \([0, 0, 0, -500259, 110375458]\) | \(192100033/38148\) | \(2749494014085316608\) | \([2]\) | \(5308416\) | \(2.2545\) | \(\Gamma_0(N)\)-optimal* |
457776.er1 | 457776er2 | \([0, 0, 0, -7574979, 8024157250]\) | \(666940371553/37026\) | \(2668626543082807296\) | \([2]\) | \(10616832\) | \(2.6010\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 457776er have rank \(0\).
Complex multiplication
The elliptic curves in class 457776er do not have complex multiplication.Modular form 457776.2.a.er
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.