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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 418968bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
418968.bg2 | 418968bg1 | \([0, 0, 0, -71415, 5913162]\) | \(54000/11\) | \(8205233775374592\) | \([2]\) | \(2162688\) | \(1.7692\) | \(\Gamma_0(N)\)-optimal |
418968.bg1 | 418968bg2 | \([0, 0, 0, -357075, -76871106]\) | \(1687500/121\) | \(361030286116482048\) | \([2]\) | \(4325376\) | \(2.1158\) |
Rank
sage: E.rank()
The elliptic curves in class 418968bg have rank \(1\).
Complex multiplication
The elliptic curves in class 418968bg do not have complex multiplication.Modular form 418968.2.a.bg
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.