Properties

Label 87360.eb
Number of curves 8
Conductor 87360
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("87360.eb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 87360.eb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
87360.eb1 87360fu8 [0, 1, 0, -2572189121, 50210557479999] [2] 21233664  
87360.eb2 87360fu6 [0, 1, 0, -160761921, 784498732479] [2, 2] 10616832  
87360.eb3 87360fu7 [0, 1, 0, -158786241, 804722188095] [2] 21233664  
87360.eb4 87360fu5 [0, 1, 0, -31769921, 68800473279] [2] 7077888  
87360.eb5 87360fu3 [0, 1, 0, -10171201, 11938220735] [2] 5308416  
87360.eb6 87360fu2 [0, 1, 0, -2649921, 292761279] [2, 2] 3538944  
87360.eb7 87360fu1 [0, 1, 0, -1646401, -809304385] [2] 1769472 \(\Gamma_0(N)\)-optimal
87360.eb8 87360fu4 [0, 1, 0, 10413759, 2333308095] [2] 7077888  

Rank

sage: E.rank()
 

The elliptic curves in class 87360.eb have rank \(1\).

Modular form 87360.2.a.eb

sage: E.q_eigenform(10)
 
\( q + q^{3} - q^{5} - q^{7} + q^{9} - q^{13} - q^{15} + 6q^{17} - 4q^{19} + O(q^{20}) \)

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.