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SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 34650du
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34650.cr2 | 34650du1 | \([1, -1, 1, 625, 627]\) | \(296740963/174636\) | \(-15913705500\) | \([2]\) | \(24576\) | \(0.64692\) | \(\Gamma_0(N)\)-optimal |
34650.cr1 | 34650du2 | \([1, -1, 1, -2525, 6927]\) | \(19530306557/11114334\) | \(1012793685750\) | \([2]\) | \(49152\) | \(0.99349\) |
Rank
sage: E.rank()
The elliptic curves in class 34650du have rank \(0\).
Complex multiplication
The elliptic curves in class 34650du do not have complex multiplication.Modular form 34650.2.a.du
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.