Properties

Label 75b
Number of curves 8
Conductor 75
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("75.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75.b7 75b1 [1, 0, 1, -1, 23] [2] 6 \(\Gamma_0(N)\)-optimal
75.b6 75b2 [1, 0, 1, -126, 523] [2, 2] 12  
75.b5 75b3 [1, 0, 1, -251, -727] [2, 2] 24  
75.b4 75b4 [1, 0, 1, -2001, 34273] [2] 24  
75.b2 75b5 [1, 0, 1, -3376, -75727] [2, 2] 48  
75.b8 75b6 [1, 0, 1, 874, -5227] [2] 48  
75.b1 75b7 [1, 0, 1, -54001, -4834477] [2] 96  
75.b3 75b8 [1, 0, 1, -2751, -104477] [4] 96  

Rank

sage: E.rank()
 

The elliptic curves in class 75b have rank \(0\).

Modular form 75.2.a.b

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.