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SageMath
E = EllipticCurve("fj1")
E.isogeny_class()
Elliptic curves in class 270480fj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 270480.fj5 | 270480fj1 | \([0, 1, 0, -329296, 79685780]\) | \(-8194759433281/965779200\) | \(-465399632284876800\) | \([2]\) | \(3538944\) | \(2.1260\) | \(\Gamma_0(N)\)-optimal |
| 270480.fj4 | 270480fj2 | \([0, 1, 0, -5409616, 4840961684]\) | \(36330796409313601/428490000\) | \(206485176360960000\) | \([2, 2]\) | \(7077888\) | \(2.4725\) | |
| 270480.fj1 | 270480fj3 | \([0, 1, 0, -86553616, 309909944084]\) | \(148809678420065817601/20700\) | \(9975129292800\) | \([2]\) | \(14155776\) | \(2.8191\) | |
| 270480.fj3 | 270480fj4 | \([0, 1, 0, -5550736, 4574922260]\) | \(39248884582600321/3935264062500\) | \(1896365595398400000000\) | \([2, 2]\) | \(14155776\) | \(2.8191\) | |
| 270480.fj6 | 270480fj5 | \([0, 1, 0, 6891344, 22177977044]\) | \(75108181893694559/484313964843750\) | \(-233386203750000000000000\) | \([2]\) | \(28311552\) | \(3.1657\) | |
| 270480.fj2 | 270480fj6 | \([0, 1, 0, -20250736, -30052397740]\) | \(1905890658841300321/293666194803750\) | \(141514891888502307840000\) | \([2]\) | \(28311552\) | \(3.1657\) |
Rank
sage: E.rank()
The elliptic curves in class 270480fj have rank \(1\).
Complex multiplication
The elliptic curves in class 270480fj do not have complex multiplication.Modular form 270480.2.a.fj
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
