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SageMath
E = EllipticCurve("gx1")
E.isogeny_class()
Elliptic curves in class 62400gx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 62400.fx4 | 62400gx1 | \([0, 1, 0, 8367, -205137]\) | \(253012016/219375\) | \(-56160000000000\) | \([2]\) | \(147456\) | \(1.3258\) | \(\Gamma_0(N)\)-optimal |
| 62400.fx3 | 62400gx2 | \([0, 1, 0, -41633, -1855137]\) | \(7793764996/3080025\) | \(3153945600000000\) | \([2, 2]\) | \(294912\) | \(1.6724\) | |
| 62400.fx2 | 62400gx3 | \([0, 1, 0, -301633, 62364863]\) | \(1481943889298/34543665\) | \(70745425920000000\) | \([4]\) | \(589824\) | \(2.0189\) | |
| 62400.fx1 | 62400gx4 | \([0, 1, 0, -581633, -170875137]\) | \(10625310339698/3855735\) | \(7896545280000000\) | \([2]\) | \(589824\) | \(2.0189\) |
Rank
sage: E.rank()
The elliptic curves in class 62400gx have rank \(0\).
Complex multiplication
The elliptic curves in class 62400gx do not have complex multiplication.Modular form 62400.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
