Properties

Label 525b
Number of curves 6
Conductor 525
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("525.d1")
sage: E.isogeny_class()

Elliptic curves in class 525b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
525.d6 525b1 [1, 1, 0, 25, 0] 2 64 \(\Gamma_0(N)\)-optimal
525.d5 525b2 [1, 1, 0, -100, -125] 4 128  
525.d2 525b3 [1, 1, 0, -1225, -17000] 4 256  
525.d3 525b4 [1, 1, 0, -975, 11250] 2 256  
525.d1 525b5 [1, 1, 0, -19600, -1064375] 2 512  
525.d4 525b6 [1, 1, 0, -850, -27125] 2 512  

Rank

sage: E.rank()

The elliptic curves in class 525b have rank \(0\).

Modular form 525.2.a.d

sage: E.q_eigenform(10)
\( q + q^{2} - q^{3} - q^{4} - q^{6} + q^{7} - 3q^{8} + q^{9} + 4q^{11} + q^{12} + 2q^{13} + q^{14} - q^{16} + 6q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

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 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.