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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 161172.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
161172.r1 | 161172l2 | \([0, 0, 0, -170786055, 859066363694]\) | \(1666315860501346000/40252707\) | \(13308180929546773248\) | \([2]\) | \(11059200\) | \(3.1896\) | |
161172.r2 | 161172l1 | \([0, 0, 0, -10686720, 13389656357]\) | \(6532108386304000/31987847133\) | \(660980479512607726032\) | \([2]\) | \(5529600\) | \(2.8430\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 161172.r have rank \(0\).
Complex multiplication
The elliptic curves in class 161172.r do not have complex multiplication.Modular form 161172.2.a.r
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.