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SageMath
E = EllipticCurve("gg1")
E.isogeny_class()
Elliptic curves in class 130050.gg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 130050.gg1 | 130050dj4 | \([1, -1, 1, -8302880, -9206287253]\) | \(8527173507/200\) | \(1484686783209375000\) | \([2]\) | \(5308416\) | \(2.5988\) | |
| 130050.gg2 | 130050dj3 | \([1, -1, 1, -499880, -154807253]\) | \(-1860867/320\) | \(-2375498853135000000\) | \([2]\) | \(2654208\) | \(2.2523\) | |
| 130050.gg3 | 130050dj2 | \([1, -1, 1, -174755, 7393997]\) | \(57960603/31250\) | \(318219903808593750\) | \([2]\) | \(1769472\) | \(2.0495\) | |
| 130050.gg4 | 130050dj1 | \([1, -1, 1, 41995, 891497]\) | \(804357/500\) | \(-5091518460937500\) | \([2]\) | \(884736\) | \(1.7030\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 130050.gg have rank \(0\).
Complex multiplication
The elliptic curves in class 130050.gg do not have complex multiplication.Modular form 130050.2.a.gg
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.
