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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 436800.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
436800.p1 | 436800p4 | \([0, -1, 0, -143365633, -660669540863]\) | \(79560762543506753209/479824800\) | \(1965362380800000000\) | \([2]\) | \(47185920\) | \(3.1182\) | |
436800.p2 | 436800p2 | \([0, -1, 0, -8965633, -10307940863]\) | \(19458380202497209/47698560000\) | \(195373301760000000000\) | \([2, 2]\) | \(23592960\) | \(2.7717\) | |
436800.p3 | 436800p3 | \([0, -1, 0, -5637633, -18058852863]\) | \(-4837870546133689/31603162500000\) | \(-129446553600000000000000\) | \([2]\) | \(47185920\) | \(3.1182\) | |
436800.p4 | 436800p1 | \([0, -1, 0, -773633, -26980863]\) | \(12501706118329/7156531200\) | \(29313151795200000000\) | \([2]\) | \(11796480\) | \(2.4251\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 436800.p have rank \(1\).
Complex multiplication
The elliptic curves in class 436800.p do not have complex multiplication.Modular form 436800.2.a.p
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.