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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 36432cb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 36432.ce6 | 36432cb1 | \([0, 0, 0, 4461, -336238]\) | \(3288008303/18259263\) | \(-54521867169792\) | \([2]\) | \(65536\) | \(1.3178\) | \(\Gamma_0(N)\)-optimal |
| 36432.ce5 | 36432cb2 | \([0, 0, 0, -53859, -4336990]\) | \(5786435182177/627352209\) | \(1873263658438656\) | \([2, 2]\) | \(131072\) | \(1.6644\) | |
| 36432.ce4 | 36432cb3 | \([0, 0, 0, -202899, 30508562]\) | \(309368403125137/44372288367\) | \(132494943107248128\) | \([2]\) | \(262144\) | \(2.0110\) | |
| 36432.ce2 | 36432cb4 | \([0, 0, 0, -837939, -295230670]\) | \(21790813729717297/304746849\) | \(909969215164416\) | \([2, 2]\) | \(262144\) | \(2.0110\) | |
| 36432.ce3 | 36432cb5 | \([0, 0, 0, -814179, -312760798]\) | \(-19989223566735457/2584262514273\) | \(-7716566519418949632\) | \([2]\) | \(524288\) | \(2.3576\) | |
| 36432.ce1 | 36432cb6 | \([0, 0, 0, -13406979, -18894896062]\) | \(89254274298475942657/17457\) | \(52126322688\) | \([2]\) | \(524288\) | \(2.3576\) |
Rank
sage: E.rank()
The elliptic curves in class 36432cb have rank \(0\).
Complex multiplication
The elliptic curves in class 36432cb do not have complex multiplication.Modular form 36432.2.a.cb
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.
