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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 162624cj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162624.bj4 | 162624cj1 | \([0, -1, 0, -3549, 529245]\) | \(-2725888/64827\) | \(-117601264585728\) | \([2]\) | \(552960\) | \(1.3806\) | \(\Gamma_0(N)\)-optimal |
| 162624.bj3 | 162624cj2 | \([0, -1, 0, -122129, 16395249]\) | \(6940769488/35721\) | \(1036811149000704\) | \([2, 2]\) | \(1105920\) | \(1.7271\) | |
| 162624.bj1 | 162624cj3 | \([0, -1, 0, -1951649, 1050074049]\) | \(7080974546692/189\) | \(21943093100544\) | \([2]\) | \(2211840\) | \(2.0737\) | |
| 162624.bj2 | 162624cj4 | \([0, -1, 0, -189889, -3756575]\) | \(6522128932/3720087\) | \(431905901498007552\) | \([2]\) | \(2211840\) | \(2.0737\) |
Rank
sage: E.rank()
The elliptic curves in class 162624cj have rank \(0\).
Complex multiplication
The elliptic curves in class 162624cj do not have complex multiplication.Modular form 162624.2.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.
