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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 33462.u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33462.u1 | 33462k4 | \([1, -1, 0, -223872, 10245860]\) | \(13060888875/7086244\) | \(673236275325939468\) | \([2]\) | \(442368\) | \(2.1118\) | |
33462.u2 | 33462k2 | \([1, -1, 0, -173172, 27780624]\) | \(4406910829875/7744\) | \(1009227840192\) | \([2]\) | \(147456\) | \(1.5625\) | |
33462.u3 | 33462k3 | \([1, -1, 0, -132612, -18428032]\) | \(2714704875/21296\) | \(2023249512624912\) | \([2]\) | \(221184\) | \(1.7653\) | |
33462.u4 | 33462k1 | \([1, -1, 0, -10932, 426960]\) | \(1108717875/45056\) | \(5871871070208\) | \([2]\) | \(73728\) | \(1.2160\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33462.u have rank \(2\).
Complex multiplication
The elliptic curves in class 33462.u do not have complex multiplication.Modular form 33462.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.