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SageMath
E = EllipticCurve("gt1")
E.isogeny_class()
Elliptic curves in class 463680.gt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.gt1 | 463680gt4 | \([0, 0, 0, -26496588, -52496825488]\) | \(10765299591712341649/20708625\) | \(3957479866368000\) | \([2]\) | \(17301504\) | \(2.6745\) | |
463680.gt2 | 463680gt2 | \([0, 0, 0, -1656588, -819689488]\) | \(2630872462131649/3645140625\) | \(696597221376000000\) | \([2, 2]\) | \(8650752\) | \(2.3279\) | |
463680.gt3 | 463680gt3 | \([0, 0, 0, -1192908, -1288377232]\) | \(-982374577874929/3183837890625\) | \(-608440896000000000000\) | \([2]\) | \(17301504\) | \(2.6745\) | |
463680.gt4 | 463680gt1 | \([0, 0, 0, -133068, -4910992]\) | \(1363569097969/734582625\) | \(140380925755392000\) | \([2]\) | \(4325376\) | \(1.9814\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 463680.gt have rank \(1\).
Complex multiplication
The elliptic curves in class 463680.gt do not have complex multiplication.Modular form 463680.2.a.gt
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.