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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 12480.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12480.a1 | 12480bp4 | \([0, -1, 0, -910081, 334370881]\) | \(2543984126301795848/909361981125\) | \(29797973397504000\) | \([2]\) | \(196608\) | \(2.1304\) | |
| 12480.a2 | 12480bp3 | \([0, -1, 0, -470081, -121345119]\) | \(350584567631475848/8259273550125\) | \(270639875690496000\) | \([2]\) | \(196608\) | \(2.1304\) | |
| 12480.a3 | 12480bp2 | \([0, -1, 0, -65081, 3637881]\) | \(7442744143086784/2927948765625\) | \(11992878144000000\) | \([2, 2]\) | \(98304\) | \(1.7838\) | |
| 12480.a4 | 12480bp1 | \([0, -1, 0, 13044, 403506]\) | \(3834800837445824/3342041015625\) | \(-213890625000000\) | \([2]\) | \(49152\) | \(1.4372\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12480.a have rank \(0\).
Complex multiplication
The elliptic curves in class 12480.a do not have complex multiplication.Modular form 12480.2.a.a
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
