Properties

Label 170.a
Number of curves $4$
Conductor $170$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 170.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
170.a1 170b3 \([1, 0, 1, -4169, -20724]\) \(8010684753304969/4456448000000\) \(4456448000000\) \([2]\) \(480\) \(1.1175\)  
170.a2 170b1 \([1, 0, 1, -2554, 49452]\) \(1841373668746009/31443200\) \(31443200\) \([6]\) \(160\) \(0.56817\) \(\Gamma_0(N)\)-optimal
170.a3 170b2 \([1, 0, 1, -2474, 52716]\) \(-1673672305534489/241375690000\) \(-241375690000\) \([6]\) \(320\) \(0.91474\)  
170.a4 170b4 \([1, 0, 1, 16311, -159988]\) \(479958568556831351/289000000000000\) \(-289000000000000\) \([2]\) \(960\) \(1.4641\)  

Rank

sage: E.rank()
 

The elliptic curves in class 170.a have rank \(0\).

Complex multiplication

The elliptic curves in class 170.a do not have complex multiplication.

Modular form 170.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - 2 q^{3} + q^{4} - q^{5} + 2 q^{6} + 2 q^{7} - q^{8} + q^{9} + q^{10} + 6 q^{11} - 2 q^{12} + 2 q^{13} - 2 q^{14} + 2 q^{15} + q^{16} + q^{17} - q^{18} + 8 q^{19} + O(q^{20})\) Copy content 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.