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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 425880cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 425880.cp3 | 425880cp1 | \([0, 0, 0, -481143, 128449802]\) | \(13674725584/945\) | \(851254490661120\) | \([2]\) | \(3538944\) | \(1.9185\) | \(\Gamma_0(N)\)-optimal |
| 425880.cp2 | 425880cp2 | \([0, 0, 0, -511563, 111286838]\) | \(4108974916/893025\) | \(3217741974699033600\) | \([2, 2]\) | \(7077888\) | \(2.2651\) | |
| 425880.cp4 | 425880cp3 | \([0, 0, 0, 1131117, 679325582]\) | \(22208984782/40516875\) | \(-291980290296764160000\) | \([2]\) | \(14155776\) | \(2.6117\) | |
| 425880.cp1 | 425880cp4 | \([0, 0, 0, -2640963, -1555181602]\) | \(282678688658/18600435\) | \(134041937117462599680\) | \([2]\) | \(14155776\) | \(2.6117\) |
Rank
sage: E.rank()
The elliptic curves in class 425880cp have rank \(1\).
Complex multiplication
The elliptic curves in class 425880cp do not have complex multiplication.Modular form 425880.2.a.cp
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.
