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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 12240v
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12240.bo3 | 12240v1 | \([0, 0, 0, -8562, 249991]\) | \(5951163357184/1129312125\) | \(13172296626000\) | \([2]\) | \(27648\) | \(1.2350\) | \(\Gamma_0(N)\)-optimal |
| 12240.bo2 | 12240v2 | \([0, 0, 0, -41367, -3010826]\) | \(41948679809104/3291890625\) | \(614345796000000\) | \([2, 2]\) | \(55296\) | \(1.5816\) | |
| 12240.bo1 | 12240v3 | \([0, 0, 0, -648867, -201177326]\) | \(40472803590982276/281883375\) | \(210424811904000\) | \([2]\) | \(110592\) | \(1.9281\) | |
| 12240.bo4 | 12240v4 | \([0, 0, 0, 41253, -13536614]\) | \(10400706415004/112060546875\) | \(-83652750000000000\) | \([4]\) | \(110592\) | \(1.9281\) |
Rank
sage: E.rank()
The elliptic curves in class 12240v have rank \(0\).
Complex multiplication
The elliptic curves in class 12240v do not have complex multiplication.Modular form 12240.2.a.v
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.
