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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 20230.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20230.r1 | 20230n1 | \([1, 1, 1, -3082191, 2079399509]\) | \(27306250652897/31360000\) | \(3718915806945920000\) | \([2]\) | \(783360\) | \(2.4762\) | \(\Gamma_0(N)\)-optimal |
20230.r2 | 20230n2 | \([1, 1, 1, -2296111, 3166390933]\) | \(-11289171456737/30012500000\) | \(-3559118643366212500000\) | \([2]\) | \(1566720\) | \(2.8228\) |
Rank
sage: E.rank()
The elliptic curves in class 20230.r have rank \(1\).
Complex multiplication
The elliptic curves in class 20230.r do not have complex multiplication.Modular form 20230.2.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.