Properties

Label 162624.fe
Number of curves $4$
Conductor $162624$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("fe1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 162624.fe

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162624.fe1 162624et4 \([0, 1, 0, -309872449, 1488601497887]\) \(14171198121996897746/4077720290568771\) \(946854983851088612102111232\) \([2]\) \(88473600\) \(3.8827\)  
162624.fe2 162624et2 \([0, 1, 0, -284104289, 1842846700671]\) \(21843440425782779332/3100814593569\) \(360007742403227891073024\) \([2, 2]\) \(44236800\) \(3.5361\)  
162624.fe3 162624et1 \([0, 1, 0, -284094609, 1842978582927]\) \(87364831012240243408/1760913\) \(51110949604442112\) \([2]\) \(22118400\) \(3.1895\) \(\Gamma_0(N)\)-optimal
162624.fe4 162624et3 \([0, 1, 0, -258491009, 2188651593951]\) \(-8226100326647904626/4152140742401883\) \(-964135564836893130668507136\) \([2]\) \(88473600\) \(3.8827\)  

Rank

sage: E.rank()
 

The elliptic curves in class 162624.fe have rank \(1\).

Complex multiplication

The elliptic curves in class 162624.fe do not have complex multiplication.

Modular form 162624.2.a.fe

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2q^{5} - q^{7} + q^{9} - 6q^{13} - 2q^{15} + 2q^{17} - 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.