Properties

Label 1638.h
Number of curves $2$
Conductor $1638$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1638.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.h1 1638h2 \([1, -1, 0, -33070464, 73207840986]\) \(-5486773802537974663600129/2635437714\) \(-1921234093506\) \([]\) \(65856\) \(2.5959\)  
1638.h2 1638h1 \([1, -1, 0, 6426, 2238516]\) \(40251338884511/2997011332224\) \(-2184821261191296\) \([]\) \(9408\) \(1.6229\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1638.h have rank \(1\).

Complex multiplication

The elliptic curves in class 1638.h do not have complex multiplication.

Modular form 1638.2.a.h

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