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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 37.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
37.b1 | 37b2 | \([0, 1, 1, -1873, -31833]\) | \(727057727488000/37\) | \(37\) | \([]\) | \(6\) | \(0.22208\) | |
37.b2 | 37b1 | \([0, 1, 1, -23, -50]\) | \(1404928000/50653\) | \(50653\) | \([3]\) | \(2\) | \(-0.32722\) | \(\Gamma_0(N)\)-optimal |
37.b3 | 37b3 | \([0, 1, 1, -3, 1]\) | \(4096000/37\) | \(37\) | \([3]\) | \(6\) | \(-0.87653\) |
Rank
sage: E.rank()
The elliptic curves in class 37.b have rank \(0\).
Complex multiplication
The elliptic curves in class 37.b do not have complex multiplication.Modular form 37.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.