Properties

Label 31713.h
Number of curves $4$
Conductor $31713$
CM no
Rank $1$
Graph

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

Elliptic curves in class 31713.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31713.h1 31713e4 \([1, 0, 1, -140807, -20328541]\) \(347873904937/395307\) \(350836417625067\) \([2]\) \(172800\) \(1.7044\)  
31713.h2 31713e2 \([1, 0, 1, -11072, -141775]\) \(169112377/88209\) \(78285812197329\) \([2, 2]\) \(86400\) \(1.3578\)  
31713.h3 31713e1 \([1, 0, 1, -6267, 188809]\) \(30664297/297\) \(263588593257\) \([2]\) \(43200\) \(1.0112\) \(\Gamma_0(N)\)-optimal
31713.h4 31713e3 \([1, 0, 1, 41783, -1093165]\) \(9090072503/5845851\) \(-5188214281077531\) \([2]\) \(172800\) \(1.7044\)  

Rank

sage: E.rank()
 

The elliptic curves in class 31713.h have rank \(1\).

Complex multiplication

The elliptic curves in class 31713.h do not have complex multiplication.

Modular form 31713.2.a.h

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.