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SageMath
sage: E = EllipticCurve("i1")
sage: E.isogeny_class()
Elliptic curves in class 1089.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
1089.i1 | 1089f2 | [1, -1, 0, -2745, -215726] | [] | 1584 | |
1089.i2 | 1089f1 | [1, -1, 0, -270, 1777] | [] | 144 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1089.i have rank \(1\).
Complex multiplication
The elliptic curves in class 1089.i do not have complex multiplication.Modular form 1089.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.