Properties

Label 32175p
Number of curves $6$
Conductor $32175$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("32175.j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 32175p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
32175.j5 32175p1 [1, -1, 1, -5405, -218028] [2] 65536 \(\Gamma_0(N)\)-optimal
32175.j4 32175p2 [1, -1, 1, -96530, -11517528] [2, 2] 131072  
32175.j3 32175p3 [1, -1, 1, -106655, -8945778] [2, 2] 262144  
32175.j1 32175p4 [1, -1, 1, -1544405, -738350778] [2] 262144  
32175.j6 32175p5 [1, -1, 1, 301720, -61217778] [2] 524288  
32175.j2 32175p6 [1, -1, 1, -677030, 207796722] [2] 524288  

Rank

sage: E.rank()
 

The elliptic curves in class 32175p have rank \(1\).

Modular form 32175.2.a.j

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{4} + 3q^{8} + q^{11} - q^{13} - q^{16} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.