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Properties

Label 174570.w
Number of curves $2$
Conductor $174570$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 174570.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
174570.w1 174570bx2 \([1, 0, 1, -3244524969, -71133871376324]\) \(2097135779881817564063/54450000\) \(98072762416660350000\) \([2]\) \(103864320\) \(3.8024\)  
174570.w2 174570bx1 \([1, 0, 1, -202774969, -1111569676324]\) \(-511936642613564063/82500000000\) \(-148595094570697500000000\) \([2]\) \(51932160\) \(3.4558\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 174570.w have rank \(1\).

Complex multiplication

The elliptic curves in class 174570.w do not have complex multiplication.

Modular form 174570.2.a.w

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.