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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 18720bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18720.x3 | 18720bn1 | \([0, 0, 0, -777, -5704]\) | \(1111934656/342225\) | \(15966849600\) | \([2, 2]\) | \(12288\) | \(0.66248\) | \(\Gamma_0(N)\)-optimal |
| 18720.x1 | 18720bn2 | \([0, 0, 0, -11307, -462706]\) | \(428320044872/73125\) | \(27293760000\) | \([2]\) | \(24576\) | \(1.0091\) | |
| 18720.x2 | 18720bn3 | \([0, 0, 0, -4827, 124706]\) | \(33324076232/1285245\) | \(479715125760\) | \([2]\) | \(24576\) | \(1.0091\) | |
| 18720.x4 | 18720bn4 | \([0, 0, 0, 2148, -38464]\) | \(367061696/426465\) | \(-1273417666560\) | \([2]\) | \(24576\) | \(1.0091\) |
Rank
sage: E.rank()
The elliptic curves in class 18720bn have rank \(0\).
Complex multiplication
The elliptic curves in class 18720bn do not have complex multiplication.Modular form 18720.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
