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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1140.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1140.d1 | 1140c4 | \([0, 1, 0, -36956, -2745900]\) | \(21804712949838544/8680921875\) | \(2222316000000\) | \([2]\) | \(3456\) | \(1.3331\) | |
| 1140.d2 | 1140c3 | \([0, 1, 0, -2661, -29736]\) | \(130287139815424/52926616125\) | \(846825858000\) | \([2]\) | \(1728\) | \(0.98648\) | |
| 1140.d3 | 1140c2 | \([0, 1, 0, -1316, 13284]\) | \(985329269584/252434475\) | \(64623225600\) | \([6]\) | \(1152\) | \(0.78375\) | |
| 1140.d4 | 1140c1 | \([0, 1, 0, -1221, 16020]\) | \(12592337649664/1315845\) | \(21053520\) | \([6]\) | \(576\) | \(0.43718\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1140.d have rank \(0\).
Complex multiplication
The elliptic curves in class 1140.d do not have complex multiplication.Modular form 1140.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
