Properties

Label 175b
Number of curves $3$
Conductor $175$
CM no
Rank $1$
Graph

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

Elliptic curves in class 175b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
175.b2 175b1 \([0, -1, 1, -33, 93]\) \(-262144/35\) \(-546875\) \([]\) \(16\) \(-0.16643\) \(\Gamma_0(N)\)-optimal
175.b3 175b2 \([0, -1, 1, 217, -282]\) \(71991296/42875\) \(-669921875\) \([]\) \(48\) \(0.38287\)  
175.b1 175b3 \([0, -1, 1, -3283, -74657]\) \(-250523582464/13671875\) \(-213623046875\) \([]\) \(144\) \(0.93218\)  

Rank

sage: E.rank()
 

The elliptic curves in class 175b have rank \(1\).

Complex multiplication

The elliptic curves in class 175b do not have complex multiplication.

Modular form 175.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2q^{4} - q^{7} - 2q^{9} - 3q^{11} + 2q^{12} - 5q^{13} + 4q^{16} - 3q^{17} + 2q^{19} + O(q^{20})\)  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 Cremona numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.