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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 20181d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20181.e6 | 20181d1 | \([1, 1, 1, 941, 2840]\) | \(103823/63\) | \(-55912731903\) | \([2]\) | \(15360\) | \(0.75148\) | \(\Gamma_0(N)\)-optimal |
| 20181.e5 | 20181d2 | \([1, 1, 1, -3864, 18216]\) | \(7189057/3969\) | \(3522502109889\) | \([2, 2]\) | \(30720\) | \(1.0981\) | |
| 20181.e3 | 20181d3 | \([1, 1, 1, -37499, -2793670]\) | \(6570725617/45927\) | \(40760381557287\) | \([2]\) | \(61440\) | \(1.4446\) | |
| 20181.e2 | 20181d4 | \([1, 1, 1, -47109, 3910266]\) | \(13027640977/21609\) | \(19178067042729\) | \([2, 2]\) | \(61440\) | \(1.4446\) | |
| 20181.e1 | 20181d5 | \([1, 1, 1, -753444, 251410050]\) | \(53297461115137/147\) | \(130463041107\) | \([2]\) | \(122880\) | \(1.7912\) | |
| 20181.e4 | 20181d6 | \([1, 1, 1, -32694, 6366582]\) | \(-4354703137/17294403\) | \(-15348846323197443\) | \([2]\) | \(122880\) | \(1.7912\) |
Rank
sage: E.rank()
The elliptic curves in class 20181d have rank \(0\).
Complex multiplication
The elliptic curves in class 20181d do not have complex multiplication.Modular form 20181.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
