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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 115920.ex
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
115920.ex1 | 115920bu2 | \([0, 0, 0, -6627, -13246]\) | \(21558430658/12425175\) | \(18550686873600\) | \([2]\) | \(229376\) | \(1.2355\) | |
115920.ex2 | 115920bu1 | \([0, 0, 0, 1653, -1654]\) | \(669136604/388815\) | \(-290248842240\) | \([2]\) | \(114688\) | \(0.88895\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 115920.ex have rank \(1\).
Complex multiplication
The elliptic curves in class 115920.ex do not have complex multiplication.Modular form 115920.2.a.ex
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.