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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 10080bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10080.c3 | 10080bn1 | \([0, 0, 0, -47253, 3952748]\) | \(250094631024064/62015625\) | \(2893401000000\) | \([2, 2]\) | \(24576\) | \(1.3787\) | \(\Gamma_0(N)\)-optimal |
| 10080.c2 | 10080bn2 | \([0, 0, 0, -52923, 2944622]\) | \(43919722445768/15380859375\) | \(5740875000000000\) | \([2]\) | \(49152\) | \(1.7253\) | |
| 10080.c1 | 10080bn3 | \([0, 0, 0, -756003, 253007498]\) | \(128025588102048008/7875\) | \(2939328000\) | \([2]\) | \(49152\) | \(1.7253\) | |
| 10080.c4 | 10080bn4 | \([0, 0, 0, -41628, 4929248]\) | \(-2671731885376/1969120125\) | \(-5879761187328000\) | \([2]\) | \(49152\) | \(1.7253\) |
Rank
sage: E.rank()
The elliptic curves in class 10080bn have rank \(1\).
Complex multiplication
The elliptic curves in class 10080bn do not have complex multiplication.Modular form 10080.2.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.
