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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 12480ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12480.bl4 | 12480ca1 | \([0, -1, 0, 335, -1775]\) | \(253012016/219375\) | \(-3594240000\) | \([2]\) | \(6144\) | \(0.52108\) | \(\Gamma_0(N)\)-optimal |
| 12480.bl3 | 12480ca2 | \([0, -1, 0, -1665, -14175]\) | \(7793764996/3080025\) | \(201852518400\) | \([2, 2]\) | \(12288\) | \(0.86766\) | |
| 12480.bl1 | 12480ca3 | \([0, -1, 0, -23265, -1357695]\) | \(10625310339698/3855735\) | \(505378897920\) | \([2]\) | \(24576\) | \(1.2142\) | |
| 12480.bl2 | 12480ca4 | \([0, -1, 0, -12065, 503745]\) | \(1481943889298/34543665\) | \(4527707258880\) | \([2]\) | \(24576\) | \(1.2142\) |
Rank
sage: E.rank()
The elliptic curves in class 12480ca have rank \(1\).
Complex multiplication
The elliptic curves in class 12480ca do not have complex multiplication.Modular form 12480.2.a.ca
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.
