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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 8050.u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8050.u1 | 8050o2 | \([1, 1, 1, -158, -1829]\) | \(-17455277065/43606528\) | \(-1090163200\) | \([]\) | \(5184\) | \(0.42202\) | |
8050.u2 | 8050o1 | \([1, 1, 1, 17, 61]\) | \(21653735/63112\) | \(-1577800\) | \([]\) | \(1728\) | \(-0.12728\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8050.u have rank \(1\).
Complex multiplication
The elliptic curves in class 8050.u do not have complex multiplication.Modular form 8050.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.