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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 172425cj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 172425.ca3 | 172425cj1 | \([1, 1, 0, -4600, 101875]\) | \(389017/57\) | \(1577796515625\) | \([2]\) | \(276480\) | \(1.0655\) | \(\Gamma_0(N)\)-optimal |
| 172425.ca2 | 172425cj2 | \([1, 1, 0, -19725, -972000]\) | \(30664297/3249\) | \(89934401390625\) | \([2, 2]\) | \(552960\) | \(1.4121\) | |
| 172425.ca4 | 172425cj3 | \([1, 1, 0, 25650, -4738125]\) | \(67419143/390963\) | \(-10822106300671875\) | \([2]\) | \(1105920\) | \(1.7587\) | |
| 172425.ca1 | 172425cj4 | \([1, 1, 0, -307100, -65631375]\) | \(115714886617/1539\) | \(42600505921875\) | \([2]\) | \(1105920\) | \(1.7587\) |
Rank
sage: E.rank()
The elliptic curves in class 172425cj have rank \(1\).
Complex multiplication
The elliptic curves in class 172425cj do not have complex multiplication.Modular form 172425.2.a.cj
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.
