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SageMath
E = EllipticCurve("fj1")
E.isogeny_class()
Elliptic curves in class 349830.fj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 349830.fj1 | 349830fj4 | \([1, -1, 1, -167918432, -837478947769]\) | \(148809678420065817601/20700\) | \(72837995852700\) | \([2]\) | \(28311552\) | \(2.9848\) | |
| 349830.fj2 | 349830fj5 | \([1, -1, 1, -39287462, 81199588799]\) | \(1905890658841300321/293666194803750\) | \(1033336090782305931903750\) | \([2]\) | \(56623104\) | \(3.3314\) | |
| 349830.fj3 | 349830fj3 | \([1, -1, 1, -10768712, -12364726201]\) | \(39248884582600321/3935264062500\) | \(13847185867809389062500\) | \([2, 2]\) | \(28311552\) | \(2.9848\) | |
| 349830.fj4 | 349830fj2 | \([1, -1, 1, -10494932, -13083562969]\) | \(36330796409313601/428490000\) | \(1507746514150890000\) | \([2, 2]\) | \(14155776\) | \(2.6382\) | |
| 349830.fj5 | 349830fj1 | \([1, -1, 1, -638852, -215464921]\) | \(-8194759433281/965779200\) | \(-3398329534503571200\) | \([2]\) | \(7077888\) | \(2.2916\) | \(\Gamma_0(N)\)-optimal |
| 349830.fj6 | 349830fj6 | \([1, -1, 1, 13369558, -59926773409]\) | \(75108181893694559/484313964843750\) | \(-1704176742159118652343750\) | \([2]\) | \(56623104\) | \(3.3314\) |
Rank
sage: E.rank()
The elliptic curves in class 349830.fj have rank \(1\).
Complex multiplication
The elliptic curves in class 349830.fj do not have complex multiplication.Modular form 349830.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
