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SageMath
E = EllipticCurve("gv1")
E.isogeny_class()
Elliptic curves in class 485100.gv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485100.gv1 | 485100gv3 | \([0, 0, 0, -11377800, -1193511375]\) | \(281370820608/161767375\) | \(93650581742191031250000\) | \([2]\) | \(35831808\) | \(3.0974\) | \(\Gamma_0(N)\)-optimal* |
485100.gv2 | 485100gv1 | \([0, 0, 0, -8143800, -8945139875]\) | \(75216478666752/326095\) | \(258962066921250000\) | \([2]\) | \(11943936\) | \(2.5481\) | \(\Gamma_0(N)\)-optimal* |
485100.gv3 | 485100gv2 | \([0, 0, 0, -8015175, -9241363250]\) | \(-4481782160112/310023175\) | \(-3939182983682100000000\) | \([2]\) | \(23887872\) | \(2.8947\) | |
485100.gv4 | 485100gv4 | \([0, 0, 0, 45345825, -9531884250]\) | \(1113258734352/648484375\) | \(-6006742852268812500000000\) | \([2]\) | \(71663616\) | \(3.4440\) |
Rank
sage: E.rank()
The elliptic curves in class 485100.gv have rank \(0\).
Complex multiplication
The elliptic curves in class 485100.gv do not have complex multiplication.Modular form 485100.2.a.gv
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.