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SageMath
sage: E = EllipticCurve("bn1")
sage: E.isogeny_class()
Elliptic curves in class 266805bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 266805.bn6 | 266805bn1 | [1, -1, 1, 1866523, 285120242124] | [2] | 44236800 | \(\Gamma_0(N)\)-optimal |
| 266805.bn5 | 266805bn2 | [1, -1, 1, -638732282, 6103294828656] | [2, 2] | 88473600 | |
| 266805.bn2 | 266805bn3 | [1, -1, 1, -10169273687, 394716838942374] | [2, 2] | 176947200 | |
| 266805.bn4 | 266805bn4 | [1, -1, 1, -1357771757, -10154762932674] | [2] | 176947200 | |
| 266805.bn1 | 266805bn5 | [1, -1, 1, -162708362312, 25261761097860774] | [2] | 353894400 | |
| 266805.bn3 | 266805bn6 | [1, -1, 1, -10118847542, 398825036805066] | [2] | 353894400 |
Rank
sage: E.rank()
The elliptic curves in class 266805bn have rank \(1\).
Complex multiplication
The elliptic curves in class 266805bn do not have complex multiplication.Modular form 266805.2.a.bn
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
