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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 19494.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19494.bp1 | 19494bd3 | \([1, -1, 1, -44471, 4797199]\) | \(-1167051/512\) | \(-4267026780931584\) | \([]\) | \(128304\) | \(1.7063\) | |
19494.bp2 | 19494bd1 | \([1, -1, 1, -1151, -14931]\) | \(-132651/2\) | \(-2540477574\) | \([]\) | \(14256\) | \(0.60771\) | \(\Gamma_0(N)\)-optimal |
19494.bp3 | 19494bd2 | \([1, -1, 1, 4264, -76301]\) | \(9261/8\) | \(-7408032605784\) | \([]\) | \(42768\) | \(1.1570\) |
Rank
sage: E.rank()
The elliptic curves in class 19494.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 19494.bp do not have complex multiplication.Modular form 19494.2.a.bp
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.