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SageMath
E = EllipticCurve("gx1")
E.isogeny_class()
Elliptic curves in class 286110.gx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286110.gx1 | 286110gx3 | \([1, -1, 1, -3028052, -1598611949]\) | \(6462919457883/1414187500\) | \(671880156873570562500\) | \([2]\) | \(13934592\) | \(2.7100\) | |
| 286110.gx2 | 286110gx1 | \([1, -1, 1, -964592, 364701459]\) | \(152298969481827/86468800\) | \(56352958911374400\) | \([2]\) | \(4644864\) | \(2.1607\) | \(\Gamma_0(N)\)-optimal |
| 286110.gx3 | 286110gx2 | \([1, -1, 1, -791192, 499814739]\) | \(-84044939142627/116825833960\) | \(-76137073961185167480\) | \([2]\) | \(9289728\) | \(2.5073\) | |
| 286110.gx4 | 286110gx4 | \([1, -1, 1, 6725698, -9783958949]\) | \(70819203762117/127995282250\) | \(-60810529238313124470750\) | \([2]\) | \(27869184\) | \(3.0566\) |
Rank
sage: E.rank()
The elliptic curves in class 286110.gx have rank \(0\).
Complex multiplication
The elliptic curves in class 286110.gx do not have complex multiplication.Modular form 286110.2.a.gx
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
