Properties

Label 4002.h
Number of curves $2$
Conductor $4002$
CM no
Rank $1$
Graph

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

Elliptic curves in class 4002.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4002.h1 4002k2 \([1, 1, 1, -92909, -10938229]\) \(88694637150489389137/6546157250904\) \(6546157250904\) \([2]\) \(16128\) \(1.5094\)  
4002.h2 4002k1 \([1, 1, 1, -5429, -195685]\) \(-17696534894747857/5921086039488\) \(-5921086039488\) \([2]\) \(8064\) \(1.1628\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 4002.h have rank \(1\).

Complex multiplication

The elliptic curves in class 4002.h do not have complex multiplication.

Modular form 4002.2.a.h

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