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Properties

Label 175a
Number of curves $2$
Conductor $175$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 175a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
175.a2 175a1 \([0, -1, 1, 2, -2]\) \(4096/7\) \(-875\) \([]\) \(8\) \(-0.75053\) \(\Gamma_0(N)\)-optimal
175.a1 175a2 \([0, -1, 1, -148, 748]\) \(-2887553024/16807\) \(-2100875\) \([5]\) \(40\) \(0.054193\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 175.2.a.a

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.