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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 80850et
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80850.fj3 | 80850et1 | \([1, 1, 1, -243062688, 1454831737281]\) | \(863913648706111516969/2486234429521920\) | \(4570359287481630720000000\) | \([4]\) | \(28901376\) | \(3.6029\) | \(\Gamma_0(N)\)-optimal |
| 80850.fj2 | 80850et2 | \([1, 1, 1, -343414688, 139016313281]\) | \(2436531580079063806249/1405478914998681600\) | \(2583643576104373305600000000\) | \([2, 2]\) | \(57802752\) | \(3.9495\) | |
| 80850.fj4 | 80850et3 | \([1, 1, 1, 1371193312, 1112913657281]\) | \(155099895405729262880471/90047655797243760000\) | \(-165531510263905173753750000000\) | \([2]\) | \(115605504\) | \(4.2960\) | |
| 80850.fj1 | 80850et4 | \([1, 1, 1, -3663654688, -85045061126719]\) | \(2958414657792917260183849/12401051653985258880\) | \(22796426969370495655830000000\) | \([2]\) | \(115605504\) | \(4.2960\) |
Rank
sage: E.rank()
The elliptic curves in class 80850et have rank \(0\).
Complex multiplication
The elliptic curves in class 80850et do not have complex multiplication.Modular form 80850.2.a.et
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.
