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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 6336.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6336.cd1 | 6336z3 | \([0, 0, 0, -17004, -853328]\) | \(5690357426/891\) | \(85136375808\) | \([2]\) | \(8192\) | \(1.1083\) | |
6336.cd2 | 6336z2 | \([0, 0, 0, -1164, -10640]\) | \(3650692/1089\) | \(52027785216\) | \([2, 2]\) | \(4096\) | \(0.76172\) | |
6336.cd3 | 6336z1 | \([0, 0, 0, -444, 3472]\) | \(810448/33\) | \(394149888\) | \([2]\) | \(2048\) | \(0.41515\) | \(\Gamma_0(N)\)-optimal |
6336.cd4 | 6336z4 | \([0, 0, 0, 3156, -71120]\) | \(36382894/43923\) | \(-4196908007424\) | \([2]\) | \(8192\) | \(1.1083\) |
Rank
sage: E.rank()
The elliptic curves in class 6336.cd have rank \(1\).
Complex multiplication
The elliptic curves in class 6336.cd do not have complex multiplication.Modular form 6336.2.a.cd
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.