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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 134640.ek
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
134640.ek1 | 134640q3 | \([0, 0, 0, -1438347, 663957434]\) | \(110211585818155849/993794670\) | \(2967454983905280\) | \([2]\) | \(1769472\) | \(2.1341\) | |
134640.ek2 | 134640q2 | \([0, 0, 0, -91947, 9876314]\) | \(28790481449449/2549240100\) | \(7611990150758400\) | \([2, 2]\) | \(884736\) | \(1.7875\) | |
134640.ek3 | 134640q1 | \([0, 0, 0, -19947, -909286]\) | \(293946977449/50490000\) | \(150762332160000\) | \([2]\) | \(442368\) | \(1.4409\) | \(\Gamma_0(N)\)-optimal |
134640.ek4 | 134640q4 | \([0, 0, 0, 102453, 46073594]\) | \(39829997144951/330164359470\) | \(-985865494747668480\) | \([4]\) | \(1769472\) | \(2.1341\) |
Rank
sage: E.rank()
The elliptic curves in class 134640.ek have rank \(0\).
Complex multiplication
The elliptic curves in class 134640.ek do not have complex multiplication.Modular form 134640.2.a.ek
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.