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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 75504.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75504.j1 | 75504bj4 | \([0, -1, 0, -13288744, 18649929520]\) | \(35765103905346817/1287\) | \(9338875932672\) | \([2]\) | \(1966080\) | \(2.4332\) | |
| 75504.j2 | 75504bj6 | \([0, -1, 0, -5825464, -5238513872]\) | \(3013001140430737/108679952667\) | \(788615846406976253952\) | \([2]\) | \(3932160\) | \(2.7798\) | |
| 75504.j3 | 75504bj3 | \([0, -1, 0, -917704, 226767664]\) | \(11779205551777/3763454409\) | \(27308806374450991104\) | \([2, 2]\) | \(1966080\) | \(2.4332\) | |
| 75504.j4 | 75504bj2 | \([0, -1, 0, -830584, 291584944]\) | \(8732907467857/1656369\) | \(12019133325348864\) | \([2, 2]\) | \(983040\) | \(2.0866\) | |
| 75504.j5 | 75504bj1 | \([0, -1, 0, -46504, 5552560]\) | \(-1532808577/938223\) | \(-6808040554917888\) | \([2]\) | \(491520\) | \(1.7400\) | \(\Gamma_0(N)\)-optimal |
| 75504.j6 | 75504bj5 | \([0, -1, 0, 2596136, 1542349360]\) | \(266679605718863/296110251723\) | \(-2148668922481252773888\) | \([2]\) | \(3932160\) | \(2.7798\) |
Rank
sage: E.rank()
The elliptic curves in class 75504.j have rank \(1\).
Complex multiplication
The elliptic curves in class 75504.j do not have complex multiplication.Modular form 75504.2.a.j
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.
