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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 121680cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121680.f2 | 121680cp1 | \([0, 0, 0, 177957, 220548042]\) | \(729/25\) | \(-21373784182656921600\) | \([2]\) | \(2875392\) | \(2.3890\) | \(\Gamma_0(N)\)-optimal |
| 121680.f1 | 121680cp2 | \([0, 0, 0, -4567563, 3588918138]\) | \(12326391/625\) | \(534344604566423040000\) | \([2]\) | \(5750784\) | \(2.7356\) |
Rank
sage: E.rank()
The elliptic curves in class 121680cp have rank \(1\).
Complex multiplication
The elliptic curves in class 121680cp do not have complex multiplication.Modular form 121680.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
