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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 12880.u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12880.u1 | 12880n3 | \([0, -1, 0, -27056, 1707200]\) | \(534774372149809/5323062500\) | \(21803264000000\) | \([2]\) | \(41472\) | \(1.3781\) | |
12880.u2 | 12880n4 | \([0, -1, 0, -7056, 4155200]\) | \(-9486391169809/1813439640250\) | \(-7427848766464000\) | \([2]\) | \(82944\) | \(1.7246\) | |
12880.u3 | 12880n1 | \([0, -1, 0, -2416, -43584]\) | \(380920459249/12622400\) | \(51701350400\) | \([2]\) | \(13824\) | \(0.82876\) | \(\Gamma_0(N)\)-optimal |
12880.u4 | 12880n2 | \([0, -1, 0, 784, -153664]\) | \(12994449551/2489452840\) | \(-10196798832640\) | \([2]\) | \(27648\) | \(1.1753\) |
Rank
sage: E.rank()
The elliptic curves in class 12880.u have rank \(1\).
Complex multiplication
The elliptic curves in class 12880.u do not have complex multiplication.Modular form 12880.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.