Properties

Label 75.c
Number of curves 2
Conductor 75
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("75.c1")
sage: E.isogeny_class()

Elliptic curves in class 75.c

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
75.c1 75a1 [0, -1, 1, -8, -7] 1 6 \(\Gamma_0(N)\)-optimal
75.c2 75a2 [0, -1, 1, 42, 443] 1 30  

Rank

sage: E.rank()

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

Modular form 75.2.a.c

sage: E.q_eigenform(10)
\( q + 2q^{2} - q^{3} + 2q^{4} - 2q^{6} - 3q^{7} + q^{9} + 2q^{11} - 2q^{12} + q^{13} - 6q^{14} - 4q^{16} + 2q^{17} + 2q^{18} - 5q^{19} + O(q^{20}) \)

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 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.