Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75.b1 75b7 \([1, 0, 1, -54001, -4834477]\) \(1114544804970241/405\) \(6328125\) \([2]\) \(96\) \(1.0956\)  
75.b2 75b5 \([1, 0, 1, -3376, -75727]\) \(272223782641/164025\) \(2562890625\) \([2, 2]\) \(48\) \(0.74901\)  
75.b3 75b8 \([1, 0, 1, -2751, -104477]\) \(-147281603041/215233605\) \(-3363025078125\) \([4]\) \(96\) \(1.0956\)  
75.b4 75b4 \([1, 0, 1, -2001, 34273]\) \(56667352321/15\) \(234375\) \([2]\) \(24\) \(0.40244\)  
75.b5 75b3 \([1, 0, 1, -251, -727]\) \(111284641/50625\) \(791015625\) \([2, 2]\) \(24\) \(0.40244\)  
75.b6 75b2 \([1, 0, 1, -126, 523]\) \(13997521/225\) \(3515625\) \([2, 2]\) \(12\) \(0.055868\)  
75.b7 75b1 \([1, 0, 1, -1, 23]\) \(-1/15\) \(-234375\) \([2]\) \(6\) \(-0.29071\) \(\Gamma_0(N)\)-optimal
75.b8 75b6 \([1, 0, 1, 874, -5227]\) \(4733169839/3515625\) \(-54931640625\) \([2]\) \(48\) \(0.74901\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 75.b do not have complex multiplication.

Modular form 75.2.a.b

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.