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SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 48576.dl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
48576.dl1 | 48576dt6 | \([0, 1, 0, -5958657, 5596501503]\) | \(89254274298475942657/17457\) | \(4576247808\) | \([2]\) | \(524288\) | \(2.1548\) | |
48576.dl2 | 48576dt4 | \([0, 1, 0, -372417, 87351615]\) | \(21790813729717297/304746849\) | \(79887557984256\) | \([2, 2]\) | \(262144\) | \(1.8083\) | |
48576.dl3 | 48576dt5 | \([0, 1, 0, -361857, 92549247]\) | \(-19989223566735457/2584262514273\) | \(-677448912541581312\) | \([2]\) | \(524288\) | \(2.1548\) | |
48576.dl4 | 48576dt3 | \([0, 1, 0, -90177, -9069633]\) | \(309368403125137/44372288367\) | \(11631929161678848\) | \([2]\) | \(262144\) | \(1.8083\) | |
48576.dl5 | 48576dt2 | \([0, 1, 0, -23937, 1277055]\) | \(5786435182177/627352209\) | \(164456617476096\) | \([2, 2]\) | \(131072\) | \(1.4617\) | |
48576.dl6 | 48576dt1 | \([0, 1, 0, 1983, 100287]\) | \(3288008303/18259263\) | \(-4786556239872\) | \([2]\) | \(65536\) | \(1.1151\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 48576.dl have rank \(1\).
Complex multiplication
The elliptic curves in class 48576.dl do not have complex multiplication.Modular form 48576.2.a.dl
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.