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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 53040.bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53040.bk1 | 53040o4 | \([0, -1, 0, -90240, 7808400]\) | \(79364416584061444/20404090514925\) | \(20893788687283200\) | \([4]\) | \(393216\) | \(1.8409\) | |
| 53040.bk2 | 53040o2 | \([0, -1, 0, -31740, -2066400]\) | \(13813960087661776/714574355625\) | \(182931035040000\) | \([2, 2]\) | \(196608\) | \(1.4943\) | |
| 53040.bk3 | 53040o1 | \([0, -1, 0, -31335, -2124558]\) | \(212670222886967296/616241925\) | \(9859870800\) | \([2]\) | \(98304\) | \(1.1477\) | \(\Gamma_0(N)\)-optimal |
| 53040.bk4 | 53040o3 | \([0, -1, 0, 20280, -8225568]\) | \(900753985478876/29018422265625\) | \(-29714864400000000\) | \([4]\) | \(393216\) | \(1.8409\) |
Rank
sage: E.rank()
The elliptic curves in class 53040.bk have rank \(0\).
Complex multiplication
The elliptic curves in class 53040.bk do not have complex multiplication.Modular form 53040.2.a.bk
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
