Properties

Label 5775c
Number of curves $6$
Conductor $5775$
CM no
Rank $1$
Graph

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

Elliptic curves in class 5775c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5775.q6 5775c1 [1, 1, 0, 875, -2891000] [2] 23040 \(\Gamma_0(N)\)-optimal
5775.q5 5775c2 [1, 1, 0, -299250, -62015625] [2, 2] 46080  
5775.q2 5775c3 [1, 1, 0, -4764375, -4004721000] [2, 2] 92160  
5775.q4 5775c4 [1, 1, 0, -636125, 102716250] [2] 92160  
5775.q1 5775c5 [1, 1, 0, -76230000, -256206911625] [2] 184320  
5775.q3 5775c6 [1, 1, 0, -4740750, -4046371875] [2] 184320  

Rank

sage: E.rank()
 

The elliptic curves in class 5775c have rank \(1\).

Complex multiplication

The elliptic curves in class 5775c do not have complex multiplication.

Modular form 5775.2.a.c

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} - q^{4} - q^{6} - q^{7} - 3q^{8} + q^{9} - q^{11} + q^{12} + 2q^{13} - q^{14} - q^{16} - 2q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.