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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 8050g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8050.l3 | 8050g1 | \([1, 1, 0, -3775, 85125]\) | \(380920459249/12622400\) | \(197225000000\) | \([2]\) | \(13824\) | \(0.94033\) | \(\Gamma_0(N)\)-optimal |
8050.l4 | 8050g2 | \([1, 1, 0, 1225, 300125]\) | \(12994449551/2489452840\) | \(-38897700625000\) | \([2]\) | \(27648\) | \(1.2869\) | |
8050.l1 | 8050g3 | \([1, 1, 0, -42275, -3334375]\) | \(534774372149809/5323062500\) | \(83172851562500\) | \([2]\) | \(41472\) | \(1.4896\) | |
8050.l2 | 8050g4 | \([1, 1, 0, -11025, -8115625]\) | \(-9486391169809/1813439640250\) | \(-28334994378906250\) | \([2]\) | \(82944\) | \(1.8362\) |
Rank
sage: E.rank()
The elliptic curves in class 8050g have rank \(0\).
Complex multiplication
The elliptic curves in class 8050g do not have complex multiplication.Modular form 8050.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.