Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("175.a1")
sage: E.isogeny_class()

Elliptic curves in class 175.a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
175.a1 175a2 [0, -1, 1, -148, 748] 5 40  
175.a2 175a1 [0, -1, 1, 2, -2] 1 8 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()

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

Modular form 175.2.a.a

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

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 LMFDB 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 LMFDB labels.