Properties

Label 10725.h
Number of curves $6$
Conductor $10725$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("10725.h1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 10725.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10725.h1 10725a4 [1, 1, 0, -171600, 27289125] [2] 32768  
10725.h2 10725a5 [1, 1, 0, -75225, -7721250] [2] 65536  
10725.h3 10725a3 [1, 1, 0, -11850, 327375] [2, 2] 32768  
10725.h4 10725a2 [1, 1, 0, -10725, 423000] [2, 2] 16384  
10725.h5 10725a1 [1, 1, 0, -600, 7875] [2] 8192 \(\Gamma_0(N)\)-optimal
10725.h6 10725a6 [1, 1, 0, 33525, 2278500] [2] 65536  

Rank

sage: E.rank()
 

The elliptic curves in class 10725.h have rank \(1\).

Modular form 10725.2.a.h

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} - q^{4} - q^{6} - 3q^{8} + q^{9} - q^{11} + q^{12} - q^{13} - q^{16} + 6q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.