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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 588.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
588.b1 | 588c2 | \([0, -1, 0, -44, 120]\) | \(109744/9\) | \(790272\) | \([2]\) | \(96\) | \(-0.12578\) | |
588.b2 | 588c1 | \([0, -1, 0, -9, -6]\) | \(16384/3\) | \(16464\) | \([2]\) | \(48\) | \(-0.47235\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 588.b have rank \(1\).
Complex multiplication
The elliptic curves in class 588.b do not have complex multiplication.Modular form 588.2.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.