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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 57222.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
57222.b1 | 57222z1 | \([1, -1, 0, -77483844, -260289446576]\) | \(595099203230897/5780865024\) | \(499759029976203487936512\) | \([2]\) | \(23396352\) | \(3.3670\) | \(\Gamma_0(N)\)-optimal |
57222.b2 | 57222z2 | \([1, -1, 0, -20886084, -632419718576]\) | \(-11655394135217/1991891886336\) | \(-172200172949886458856999168\) | \([2]\) | \(46792704\) | \(3.7136\) |
Rank
sage: E.rank()
The elliptic curves in class 57222.b have rank \(1\).
Complex multiplication
The elliptic curves in class 57222.b do not have complex multiplication.Modular form 57222.2.a.b
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.