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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 1089f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1089.i2 | 1089f1 | \([1, -1, 0, -270, 1777]\) | \(-24729001\) | \(-88209\) | \([]\) | \(144\) | \(0.031345\) | \(\Gamma_0(N)\)-optimal |
1089.i1 | 1089f2 | \([1, -1, 0, -2745, -215726]\) | \(-121\) | \(-18908382534129\) | \([]\) | \(1584\) | \(1.2303\) |
Rank
sage: E.rank()
The elliptic curves in class 1089f have rank \(1\).
Complex multiplication
The elliptic curves in class 1089f do not have complex multiplication.Modular form 1089.2.a.f
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 Cremona 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 Cremona labels.