Properties

Label 176610do
Number of curves $8$
Conductor $176610$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("176610.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 176610do

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
176610.f7 176610do1 [1, 1, 0, -34498, -882188] [2] 1161216 \(\Gamma_0(N)\)-optimal
176610.f5 176610do2 [1, 1, 0, -303618, 63652788] [2, 2] 2322432  
176610.f4 176610do3 [1, 1, 0, -2254738, -1304082332] [2] 3483648  
176610.f2 176610do4 [1, 1, 0, -4845018, 4102773948] [2] 4644864  
176610.f6 176610do5 [1, 1, 0, -68138, 160152492] [2] 4644864  
176610.f3 176610do6 [1, 1, 0, -2271558, -1283659488] [2, 2] 6967296  
176610.f1 176610do7 [1, 1, 0, -5425308, 3057792762] [2] 13934592  
176610.f8 176610do8 [1, 1, 0, 613072, -4317713322] [2] 13934592  

Rank

sage: E.rank()
 

The elliptic curves in class 176610do have rank \(0\).

Modular form 176610.2.a.f

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + q^{10} - q^{12} + 2q^{13} - q^{14} + q^{15} + q^{16} + 6q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.