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SageMath
E = EllipticCurve("da1")
E.isogeny_class()
Elliptic curves in class 31680.da
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31680.da1 | 31680bf4 | \([0, 0, 0, -1900812, -1008688016]\) | \(15897679904620804/2475\) | \(118244966400\) | \([2]\) | \(262144\) | \(1.9713\) | |
| 31680.da2 | 31680bf6 | \([0, 0, 0, -1008012, 382069744]\) | \(1185450336504002/26043266205\) | \(2488472838267863040\) | \([2]\) | \(524288\) | \(2.3178\) | |
| 31680.da3 | 31680bf3 | \([0, 0, 0, -136812, -10667216]\) | \(5927735656804/2401490025\) | \(114732972652953600\) | \([2, 2]\) | \(262144\) | \(1.9713\) | |
| 31680.da4 | 31680bf2 | \([0, 0, 0, -118812, -15757616]\) | \(15529488955216/6125625\) | \(73164072960000\) | \([2, 2]\) | \(131072\) | \(1.6247\) | |
| 31680.da5 | 31680bf1 | \([0, 0, 0, -6312, -322616]\) | \(-37256083456/38671875\) | \(-28868400000000\) | \([2]\) | \(65536\) | \(1.2781\) | \(\Gamma_0(N)\)-optimal |
| 31680.da6 | 31680bf5 | \([0, 0, 0, 446388, -77618576]\) | \(102949393183198/86815346805\) | \(-8295335568453795840\) | \([2]\) | \(524288\) | \(2.3178\) |
Rank
sage: E.rank()
The elliptic curves in class 31680.da have rank \(1\).
Complex multiplication
The elliptic curves in class 31680.da do not have complex multiplication.Modular form 31680.2.a.da
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
