Show commands:
SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 85176.bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85176.bw1 | 85176cc4 | \([0, 0, 0, -20673939, 36181193438]\) | \(271210066309732/51597\) | \(185913980760388608\) | \([2]\) | \(2752512\) | \(2.7058\) | |
| 85176.bw2 | 85176cc3 | \([0, 0, 0, -2482779, -626166970]\) | \(469732169092/224827239\) | \(810096071202924420096\) | \([2]\) | \(2752512\) | \(2.7058\) | |
| 85176.bw3 | 85176cc2 | \([0, 0, 0, -1296399, 561399410]\) | \(267492843088/3651921\) | \(3289644604010209536\) | \([2, 2]\) | \(1376256\) | \(2.3592\) | |
| 85176.bw4 | 85176cc1 | \([0, 0, 0, -11154, 23395853]\) | \(-2725888/4198467\) | \(-236373272992230192\) | \([2]\) | \(688128\) | \(2.0127\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 85176.bw have rank \(0\).
Complex multiplication
The elliptic curves in class 85176.bw do not have complex multiplication.Modular form 85176.2.a.bw
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
