Properties

Label 48576.dl
Number of curves $6$
Conductor $48576$
CM no
Rank $1$
Graph

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Show commands: 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)
 
\(q + q^{3} + 2 q^{5} + q^{9} + q^{11} + 2 q^{13} + 2 q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.