Show commands:
SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 286110.ei
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286110.ei1 | 286110ei4 | \([1, -1, 1, -31130123, -66844980169]\) | \(189602977175292169/1402500\) | \(24678793640902500\) | \([2]\) | \(18874368\) | \(2.7404\) | |
| 286110.ei2 | 286110ei3 | \([1, -1, 1, -2727203, -127831993]\) | \(127483771761289/73369857660\) | \(1291037131303764405660\) | \([2]\) | \(18874368\) | \(2.7404\) | |
| 286110.ei3 | 286110ei2 | \([1, -1, 1, -1946903, -1042655713]\) | \(46380496070089/125888400\) | \(2215168517207408400\) | \([2, 2]\) | \(9437184\) | \(2.3938\) | |
| 286110.ei4 | 286110ei1 | \([1, -1, 1, -74183, -29139649]\) | \(-2565726409/19388160\) | \(-341159643291836160\) | \([2]\) | \(4718592\) | \(2.0473\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 286110.ei have rank \(1\).
Complex multiplication
The elliptic curves in class 286110.ei do not have complex multiplication.Modular form 286110.2.a.ei
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
