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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 291312bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 291312.bw3 | 291312bw1 | \([0, 0, 0, -310386, 66556411]\) | \(11745974272/357\) | \(100509995919312\) | \([2]\) | \(1474560\) | \(1.7846\) | \(\Gamma_0(N)\)-optimal |
| 291312.bw2 | 291312bw2 | \([0, 0, 0, -323391, 60675550]\) | \(830321872/127449\) | \(574113096691110144\) | \([2, 2]\) | \(2949120\) | \(2.1312\) | |
| 291312.bw4 | 291312bw3 | \([0, 0, 0, 560949, 334290346]\) | \(1083360092/3306177\) | \(-59572676621359899648\) | \([2]\) | \(5898240\) | \(2.4778\) | |
| 291312.bw1 | 291312bw4 | \([0, 0, 0, -1415811, -589314350]\) | \(17418812548/1753941\) | \(31603559036901110784\) | \([2]\) | \(5898240\) | \(2.4778\) |
Rank
sage: E.rank()
The elliptic curves in class 291312bw have rank \(1\).
Complex multiplication
The elliptic curves in class 291312bw do not have complex multiplication.Modular form 291312.2.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 Cremona 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 Cremona labels.
