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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 2166.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2166.a1 | 2166b3 | \([1, 1, 0, -154515, 23313369]\) | \(8671983378625/82308\) | \(3872252373348\) | \([2]\) | \(12960\) | \(1.5777\) | |
2166.a2 | 2166b4 | \([1, 1, 0, -150905, 24459183]\) | \(-8078253774625/846825858\) | \(-39839668543190898\) | \([2]\) | \(25920\) | \(1.9243\) | |
2166.a3 | 2166b1 | \([1, 1, 0, -2895, -5787]\) | \(57066625/32832\) | \(1544610364992\) | \([2]\) | \(4320\) | \(1.0284\) | \(\Gamma_0(N)\)-optimal |
2166.a4 | 2166b2 | \([1, 1, 0, 11545, -31779]\) | \(3616805375/2105352\) | \(-99048139655112\) | \([2]\) | \(8640\) | \(1.3750\) |
Rank
sage: E.rank()
The elliptic curves in class 2166.a have rank \(0\).
Complex multiplication
The elliptic curves in class 2166.a do not have complex multiplication.Modular form 2166.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.