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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 33813.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33813.i1 | 33813n6 | \([1, -1, 0, -52472628, 146291618079]\) | \(908031902324522977/161726530797\) | \(2845786580961301907397\) | \([2]\) | \(2359296\) | \(3.1211\) | |
| 33813.i2 | 33813n4 | \([1, -1, 0, -3612843, 1793689920]\) | \(296380748763217/92608836489\) | \(1629571739776194370689\) | \([2, 2]\) | \(1179648\) | \(2.7745\) | |
| 33813.i3 | 33813n2 | \([1, -1, 0, -1414998, -626137425]\) | \(17806161424897/668584449\) | \(11764604383877006649\) | \([2, 2]\) | \(589824\) | \(2.4279\) | |
| 33813.i4 | 33813n1 | \([1, -1, 0, -1401993, -638598816]\) | \(17319700013617/25857\) | \(454987213670457\) | \([2]\) | \(294912\) | \(2.0813\) | \(\Gamma_0(N)\)-optimal |
| 33813.i5 | 33813n3 | \([1, -1, 0, 574767, -2248591806]\) | \(1193377118543/124806800313\) | \(-2196136377829484881713\) | \([2]\) | \(1179648\) | \(2.7745\) | |
| 33813.i6 | 33813n5 | \([1, -1, 0, 10081422, 12138337701]\) | \(6439735268725823/7345472585373\) | \(-129253049646578850934773\) | \([2]\) | \(2359296\) | \(3.1211\) |
Rank
sage: E.rank()
The elliptic curves in class 33813.i have rank \(1\).
Complex multiplication
The elliptic curves in class 33813.i do not have complex multiplication.Modular form 33813.2.a.i
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.
