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SageMath
E = EllipticCurve("jn1")
E.isogeny_class()
Elliptic curves in class 159936.jn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
159936.jn1 | 159936hf3 | \([0, 1, 0, -249377, 47848863]\) | \(444893916104/9639\) | \(37159525122048\) | \([4]\) | \(688128\) | \(1.7202\) | |
159936.jn2 | 159936hf2 | \([0, 1, 0, -16137, 687735]\) | \(964430272/127449\) | \(61416437354496\) | \([2, 2]\) | \(344064\) | \(1.3737\) | |
159936.jn3 | 159936hf1 | \([0, 1, 0, -4132, -92590]\) | \(1036433728/122451\) | \(921999212736\) | \([2]\) | \(172032\) | \(1.0271\) | \(\Gamma_0(N)\)-optimal |
159936.jn4 | 159936hf4 | \([0, 1, 0, 25023, 3659487]\) | \(449455096/1753941\) | \(-6761657293504512\) | \([2]\) | \(688128\) | \(1.7202\) |
Rank
sage: E.rank()
The elliptic curves in class 159936.jn have rank \(0\).
Complex multiplication
The elliptic curves in class 159936.jn do not have complex multiplication.Modular form 159936.2.a.jn
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.