Properties

Label 177450x
Number of curves $6$
Conductor $177450$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("177450.jl1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 177450x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177450.jl6 177450x1 [1, 0, 0, 42162, -4659708] [2] 1769472 \(\Gamma_0(N)\)-optimal
177450.jl5 177450x2 [1, 0, 0, -295838, -48261708] [2, 2] 3538944  
177450.jl4 177450x3 [1, 0, 0, -1563338, 710970792] [2] 7077888  
177450.jl2 177450x4 [1, 0, 0, -4436338, -3596670208] [2, 2] 7077888  
177450.jl3 177450x5 [1, 0, 0, -4140588, -4096783458] [2] 14155776  
177450.jl1 177450x6 [1, 0, 0, -70980088, -230178138958] [2] 14155776  

Rank

sage: E.rank()
 

The elliptic curves in class 177450x have rank \(0\).

Modular form 177450.2.a.jl

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.