Properties

Label 32368bd
Number of curves 6
Conductor 32368
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("32368.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 32368bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
32368.f5 32368bd1 [0, 1, 0, -2408, -92876] [2] 36864 \(\Gamma_0(N)\)-optimal
32368.f4 32368bd2 [0, 1, 0, -48648, -4143500] [2] 73728  
32368.f6 32368bd3 [0, 1, 0, 20712, 1932436] [2] 110592  
32368.f3 32368bd4 [0, 1, 0, -164248, 20946324] [2] 221184  
32368.f2 32368bd5 [0, 1, 0, -788488, 270004212] [2] 331776  
32368.f1 32368bd6 [0, 1, 0, -12625928, 17263833076] [2] 663552  

Rank

sage: E.rank()
 

The elliptic curves in class 32368bd have rank \(1\).

Modular form 32368.2.a.f

sage: E.q_eigenform(10)
 
\( q - 2q^{3} + q^{7} + q^{9} - 4q^{13} - 2q^{19} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.