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SageMath
E = EllipticCurve("gy1")
E.isogeny_class()
Elliptic curves in class 476850.gy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
476850.gy1 | 476850gy3 | \([1, 1, 1, -2723156838, 53074301420781]\) | \(5921450764096952391481/200074809015963750\) | \(75458117309135111205058593750\) | \([2]\) | \(509607936\) | \(4.3135\) | \(\Gamma_0(N)\)-optimal* |
476850.gy2 | 476850gy2 | \([1, 1, 1, -420188088, -2169312954219]\) | \(21754112339458491481/7199734626562500\) | \(2715376427036587133789062500\) | \([2, 2]\) | \(254803968\) | \(3.9669\) | \(\Gamma_0(N)\)-optimal* |
476850.gy3 | 476850gy1 | \([1, 1, 1, -378427588, -2833137862219]\) | \(15891267085572193561/3334993530000\) | \(1257791194452008906250000\) | \([2]\) | \(127401984\) | \(3.6203\) | \(\Gamma_0(N)\)-optimal* |
476850.gy4 | 476850gy4 | \([1, 1, 1, 1214612662, -14927298007219]\) | \(525440531549759128199/559322204589843750\) | \(-210948098539366722106933593750\) | \([2]\) | \(509607936\) | \(4.3135\) |
Rank
sage: E.rank()
The elliptic curves in class 476850.gy have rank \(0\).
Complex multiplication
The elliptic curves in class 476850.gy do not have complex multiplication.Modular form 476850.2.a.gy
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.