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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 525a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
525.a3 | 525a1 | \([1, 1, 1, -63, 156]\) | \(1771561/105\) | \(1640625\) | \([4]\) | \(96\) | \(-0.054390\) | \(\Gamma_0(N)\)-optimal |
525.a2 | 525a2 | \([1, 1, 1, -188, -844]\) | \(47045881/11025\) | \(172265625\) | \([2, 2]\) | \(192\) | \(0.29218\) | |
525.a1 | 525a3 | \([1, 1, 1, -2813, -58594]\) | \(157551496201/13125\) | \(205078125\) | \([2]\) | \(384\) | \(0.63876\) | |
525.a4 | 525a4 | \([1, 1, 1, 437, -4594]\) | \(590589719/972405\) | \(-15193828125\) | \([2]\) | \(384\) | \(0.63876\) |
Rank
sage: E.rank()
The elliptic curves in class 525a have rank \(1\).
Complex multiplication
The elliptic curves in class 525a do not have complex multiplication.Modular form 525.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.