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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 3971.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3971.b1 | 3971b3 | \([0, 1, 1, -2823140, 1824832093]\) | \(-52893159101157376/11\) | \(-517504691\) | \([]\) | \(36000\) | \(1.9689\) | |
3971.b2 | 3971b2 | \([0, 1, 1, -3730, 157593]\) | \(-122023936/161051\) | \(-7576786180931\) | \([]\) | \(7200\) | \(1.1642\) | |
3971.b3 | 3971b1 | \([0, 1, 1, -120, -1247]\) | \(-4096/11\) | \(-517504691\) | \([]\) | \(1440\) | \(0.35949\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3971.b have rank \(1\).
Complex multiplication
The elliptic curves in class 3971.b do not have complex multiplication.Modular form 3971.2.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.
\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.