Show commands:
SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 9408.bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9408.bw1 | 9408db4 | \([0, 1, 0, -4214849, 3329185215]\) | \(268498407453697/252\) | \(7771926822912\) | \([2]\) | \(147456\) | \(2.2018\) | |
| 9408.bw2 | 9408db5 | \([0, 1, 0, -2866369, -1850890945]\) | \(84448510979617/933897762\) | \(28802321691846377472\) | \([2]\) | \(294912\) | \(2.5484\) | |
| 9408.bw3 | 9408db3 | \([0, 1, 0, -326209, 25271231]\) | \(124475734657/63011844\) | \(1943346986288676864\) | \([2, 2]\) | \(147456\) | \(2.2018\) | |
| 9408.bw4 | 9408db2 | \([0, 1, 0, -263489, 51927231]\) | \(65597103937/63504\) | \(1958525559373824\) | \([2, 2]\) | \(73728\) | \(1.8552\) | |
| 9408.bw5 | 9408db1 | \([0, 1, 0, -12609, 1199295]\) | \(-7189057/16128\) | \(-497403316666368\) | \([2]\) | \(36864\) | \(1.5086\) | \(\Gamma_0(N)\)-optimal |
| 9408.bw6 | 9408db6 | \([0, 1, 0, 1210431, 196452927]\) | \(6359387729183/4218578658\) | \(-130105097724898050048\) | \([2]\) | \(294912\) | \(2.5484\) |
Rank
sage: E.rank()
The elliptic curves in class 9408.bw have rank \(0\).
Complex multiplication
The elliptic curves in class 9408.bw do not have complex multiplication.Modular form 9408.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
