Properties

Label 5577.g
Number of curves $6$
Conductor $5577$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("5577.g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 5577.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5577.g1 5577g4 [1, 0, 1, -1160020, 480793679] [2] 43008  
5577.g2 5577g5 [1, 0, 1, -508525, -135200167] [2] 86016  
5577.g3 5577g3 [1, 0, 1, -80110, 5834051] [2, 2] 43008  
5577.g4 5577g2 [1, 0, 1, -72505, 7507151] [2, 2] 21504  
5577.g5 5577g1 [1, 0, 1, -4060, 142469] [2] 10752 \(\Gamma_0(N)\)-optimal
5577.g6 5577g6 [1, 0, 1, 226625, 39820289] [2] 86016  

Rank

sage: E.rank()
 

The elliptic curves in class 5577.g have rank \(1\).

Modular form 5577.2.a.g

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} - q^{4} + 2q^{5} + q^{6} - 3q^{8} + q^{9} + 2q^{10} + q^{11} - q^{12} + 2q^{15} - q^{16} - 6q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.