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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 239343.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 239343.f1 | 239343f5 | \([1, 0, 1, -7282822, 7563031751]\) | \(908031902324522977/161726530797\) | \(7608567122418497157\) | \([2]\) | \(7077888\) | \(2.6274\) | |
| 239343.f2 | 239343f3 | \([1, 0, 1, -501437, 92658035]\) | \(296380748763217/92608836489\) | \(4356864301009951809\) | \([2, 2]\) | \(3538944\) | \(2.2808\) | |
| 239343.f3 | 239343f2 | \([1, 0, 1, -196392, -32410415]\) | \(17806161424897/668584449\) | \(31454144426104569\) | \([2, 2]\) | \(1769472\) | \(1.9342\) | |
| 239343.f4 | 239343f1 | \([1, 0, 1, -194587, -33054439]\) | \(17319700013617/25857\) | \(1216465345017\) | \([2]\) | \(884736\) | \(1.5877\) | \(\Gamma_0(N)\)-optimal |
| 239343.f5 | 239343f4 | \([1, 0, 1, 79773, -116254109]\) | \(1193377118543/124806800313\) | \(-5871645875516160753\) | \([2]\) | \(3538944\) | \(2.2808\) | |
| 239343.f6 | 239343f6 | \([1, 0, 1, 1399228, 627885299]\) | \(6439735268725823/7345472585373\) | \(-345574229140220498613\) | \([2]\) | \(7077888\) | \(2.6274\) |
Rank
sage: E.rank()
The elliptic curves in class 239343.f have rank \(0\).
Complex multiplication
The elliptic curves in class 239343.f do not have complex multiplication.Modular form 239343.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
