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Properties

Label 1690.h
Number of curves $2$
Conductor $1690$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1690.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1690.h1 1690g2 \([1, -1, 1, -76927, -8857689]\) \(-1762712152495281/171798691840\) \(-4906742437642240\) \([]\) \(11760\) \(1.7518\)  
1690.h2 1690g1 \([1, -1, 1, -877, 16501]\) \(-2609064081/2500000\) \(-71402500000\) \([]\) \(1680\) \(0.77883\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1690.h have rank \(1\).

Complex multiplication

The elliptic curves in class 1690.h do not have complex multiplication.

Modular form 1690.2.a.h

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.