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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 42432.cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42432.cf1 | 42432s6 | \([0, 1, 0, -1291137, -565029153]\) | \(908031902324522977/161726530797\) | \(42395639689248768\) | \([2]\) | \(524288\) | \(2.1949\) | |
| 42432.cf2 | 42432s4 | \([0, 1, 0, -88897, -6949345]\) | \(296380748763217/92608836489\) | \(24276850832572416\) | \([2, 2]\) | \(262144\) | \(1.8483\) | |
| 42432.cf3 | 42432s2 | \([0, 1, 0, -34817, 2406495]\) | \(17806161424897/668584449\) | \(175265401798656\) | \([2, 2]\) | \(131072\) | \(1.5017\) | |
| 42432.cf4 | 42432s1 | \([0, 1, 0, -34497, 2454687]\) | \(17319700013617/25857\) | \(6778257408\) | \([2]\) | \(65536\) | \(1.1552\) | \(\Gamma_0(N)\)-optimal |
| 42432.cf5 | 42432s3 | \([0, 1, 0, 14143, 8683167]\) | \(1193377118543/124806800313\) | \(-32717353861251072\) | \([2]\) | \(262144\) | \(1.8483\) | |
| 42432.cf6 | 42432s5 | \([0, 1, 0, 248063, -46778017]\) | \(6439735268725823/7345472585373\) | \(-1925571565420019712\) | \([2]\) | \(524288\) | \(2.1949\) |
Rank
sage: E.rank()
The elliptic curves in class 42432.cf have rank \(1\).
Complex multiplication
The elliptic curves in class 42432.cf do not have complex multiplication.Modular form 42432.2.a.cf
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.
