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SageMath
E = EllipticCurve("nn1")
E.isogeny_class()
Elliptic curves in class 479808.nn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
479808.nn1 | 479808nn4 | \([0, 0, 0, -960204, -329142800]\) | \(17418812548/1753941\) | \(9858496333929578496\) | \([2]\) | \(7864320\) | \(2.3807\) | |
479808.nn2 | 479808nn2 | \([0, 0, 0, -219324, 33888400]\) | \(830321872/127449\) | \(179090331325710336\) | \([2, 2]\) | \(3932160\) | \(2.0341\) | |
479808.nn3 | 479808nn1 | \([0, 0, 0, -210504, 37172968]\) | \(11745974272/357\) | \(31353349321728\) | \([2]\) | \(1966080\) | \(1.6876\) | \(\Gamma_0(N)\)-optimal* |
479808.nn4 | 479808nn3 | \([0, 0, 0, 380436, 186707248]\) | \(1083360092/3306177\) | \(-18583255556385472512\) | \([2]\) | \(7864320\) | \(2.3807\) |
Rank
sage: E.rank()
The elliptic curves in class 479808.nn have rank \(0\).
Complex multiplication
The elliptic curves in class 479808.nn do not have complex multiplication.Modular form 479808.2.a.nn
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.