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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 466578.do
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
466578.do1 | 466578do2 | \([1, -1, 1, -21739619, 39023916859]\) | \(-67645179/8\) | \(-134379374640662566296\) | \([]\) | \(35925120\) | \(2.8878\) | \(\Gamma_0(N)\)-optimal* |
466578.do2 | 466578do1 | \([1, -1, 1, 34021, 165227339]\) | \(189/512\) | \(-11797366223597262336\) | \([]\) | \(11975040\) | \(2.3385\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 466578.do have rank \(1\).
Complex multiplication
The elliptic curves in class 466578.do do not have complex multiplication.Modular form 466578.2.a.do
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.