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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 267696bj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 267696.bj5 | 267696bj1 | \([0, 0, 0, -584571, 246771434]\) | \(-1532808577/938223\) | \(-13522396478163775488\) | \([2]\) | \(5505024\) | \(2.3729\) | \(\Gamma_0(N)\)-optimal |
| 267696.bj4 | 267696bj2 | \([0, 0, 0, -10440651, 12982798010]\) | \(8732907467857/1656369\) | \(23872872794782961664\) | \([2, 2]\) | \(11010048\) | \(2.7195\) | |
| 267696.bj1 | 267696bj3 | \([0, 0, 0, -167042811, 830978520554]\) | \(35765103905346817/1287\) | \(18549240710787072\) | \([2]\) | \(22020096\) | \(3.0660\) | |
| 267696.bj3 | 267696bj4 | \([0, 0, 0, -11535771, 10092776330]\) | \(11779205551777/3763454409\) | \(54241819531168531451904\) | \([2, 2]\) | \(22020096\) | \(3.0660\) | |
| 267696.bj6 | 267696bj5 | \([0, 0, 0, 32634069, 68776825754]\) | \(266679605718863/296110251723\) | \(-4267770268952353792831488\) | \([2]\) | \(44040192\) | \(3.4126\) | |
| 267696.bj2 | 267696bj6 | \([0, 0, 0, -73227531, -233552660614]\) | \(3013001140430737/108679952667\) | \(1566379644488833272164352\) | \([2]\) | \(44040192\) | \(3.4126\) |
Rank
sage: E.rank()
The elliptic curves in class 267696bj have rank \(0\).
Complex multiplication
The elliptic curves in class 267696bj do not have complex multiplication.Modular form 267696.2.a.bj
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.
