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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 58.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
58.b1 | 58b2 | \([1, 1, 1, -455, -3951]\) | \(-10418796526321/82044596\) | \(-82044596\) | \([]\) | \(20\) | \(0.34716\) | |
58.b2 | 58b1 | \([1, 1, 1, 5, 9]\) | \(13651919/29696\) | \(-29696\) | \([5]\) | \(4\) | \(-0.45755\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 58.b have rank \(0\).
Complex multiplication
The elliptic curves in class 58.b do not have complex multiplication.Modular form 58.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.