Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75.c1 75a1 \([0, -1, 1, -8, -7]\) \(-102400/3\) \(-1875\) \([]\) \(6\) \(-0.59310\) \(\Gamma_0(N)\)-optimal
75.c2 75a2 \([0, -1, 1, 42, 443]\) \(20480/243\) \(-94921875\) \([]\) \(30\) \(0.21162\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 75.2.a.c

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.