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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 12480cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12480.bs4 | 12480cp1 | \([0, 1, 0, -9181, 282419]\) | \(83587439220736/13990184325\) | \(14325948748800\) | \([2]\) | \(24576\) | \(1.2455\) | \(\Gamma_0(N)\)-optimal |
| 12480.bs2 | 12480cp2 | \([0, 1, 0, -140401, 20201615]\) | \(18681746265374416/693005625\) | \(11354204160000\) | \([2, 2]\) | \(49152\) | \(1.5921\) | |
| 12480.bs1 | 12480cp3 | \([0, 1, 0, -2246401, 1295174015]\) | \(19129597231400697604/26325\) | \(1725235200\) | \([2]\) | \(98304\) | \(1.9387\) | |
| 12480.bs3 | 12480cp4 | \([0, 1, 0, -133921, 22157279]\) | \(-4053153720264484/903687890625\) | \(-59224089600000000\) | \([2]\) | \(98304\) | \(1.9387\) |
Rank
sage: E.rank()
The elliptic curves in class 12480cp have rank \(1\).
Complex multiplication
The elliptic curves in class 12480cp do not have complex multiplication.Modular form 12480.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
