Properties

Label 47040.gp
Number of curves 8
Conductor 47040
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("47040.gp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 47040.gp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.gp1 47040dc8 [0, 1, 0, -1101465185, 14069962595583] [2] 10616832  
47040.gp2 47040dc6 [0, 1, 0, -68843105, 219815685375] [2, 2] 5308416  
47040.gp3 47040dc7 [0, 1, 0, -63825505, 253221862655] [2] 10616832  
47040.gp4 47040dc5 [0, 1, 0, -13665185, 19096955583] [2] 3538944  
47040.gp5 47040dc3 [0, 1, 0, -4617825, 2901224703] [2] 2654208  
47040.gp6 47040dc2 [0, 1, 0, -1811105, -493097025] [2, 2] 1769472  
47040.gp7 47040dc1 [0, 1, 0, -1560225, -750349377] [2] 884736 \(\Gamma_0(N)\)-optimal
47040.gp8 47040dc4 [0, 1, 0, 6028895, -3614985025] [2] 3538944  

Rank

sage: E.rank()
 

The elliptic curves in class 47040.gp have rank \(0\).

Modular form 47040.2.a.gp

sage: E.q_eigenform(10)
 
\( q + q^{3} + q^{5} + q^{9} + 2q^{13} + q^{15} + 6q^{17} + 8q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.