Properties

Label 1710.h
Number of curves $2$
Conductor $1710$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("h1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1710.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1710.h1 1710b2 \([1, -1, 0, -399, 1805]\) \(260549802603/104256800\) \(2814933600\) \([2]\) \(1280\) \(0.51130\)  
1710.h2 1710b1 \([1, -1, 0, 81, 173]\) \(2161700757/1848320\) \(-49904640\) \([2]\) \(640\) \(0.16473\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1710.h have rank \(1\).

Complex multiplication

The elliptic curves in class 1710.h do not have complex multiplication.

Modular form 1710.2.a.h

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