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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 74256.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
74256.cd1 | 74256dd6 | \([0, 1, 0, -649264, 201002900]\) | \(7389727131216686257/6115533215337\) | \(25049224050020352\) | \([4]\) | \(786432\) | \(2.0750\) | |
74256.cd2 | 74256dd4 | \([0, 1, 0, -49504, 1642676]\) | \(3275619238041697/1605271262049\) | \(6575191089352704\) | \([2, 4]\) | \(393216\) | \(1.7285\) | |
74256.cd3 | 74256dd2 | \([0, 1, 0, -26384, -1640364]\) | \(495909170514577/6224736609\) | \(25496521150464\) | \([2, 2]\) | \(196608\) | \(1.3819\) | |
74256.cd4 | 74256dd1 | \([0, 1, 0, -26304, -1650828]\) | \(491411892194497/78897\) | \(323162112\) | \([2]\) | \(98304\) | \(1.0353\) | \(\Gamma_0(N)\)-optimal |
74256.cd5 | 74256dd3 | \([0, 1, 0, -4544, -4252428]\) | \(-2533811507137/1904381781393\) | \(-7800347776585728\) | \([2]\) | \(393216\) | \(1.7285\) | |
74256.cd6 | 74256dd5 | \([0, 1, 0, 180336, 12766932]\) | \(158346567380527343/108665074944153\) | \(-445092146971250688\) | \([4]\) | \(786432\) | \(2.0750\) |
Rank
sage: E.rank()
The elliptic curves in class 74256.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 74256.cd do not have complex multiplication.Modular form 74256.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.