Properties

Label 55506.a
Number of curves $4$
Conductor $55506$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 55506.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
55506.a1 55506d3 \([1, 1, 0, -8464682, -9482561760]\) \(112763292123580561/1932612\) \(1149562688044452\) \([2]\) \(2240000\) \(2.4315\)  
55506.a2 55506d4 \([1, 1, 0, -8456272, -9502333670]\) \(-112427521449300721/466873642818\) \(-277707330708370558578\) \([2]\) \(4480000\) \(2.7781\)  
55506.a3 55506d1 \([1, 1, 0, -37862, 2051220]\) \(10091699281/2737152\) \(1628121842721792\) \([2]\) \(448000\) \(1.6268\) \(\Gamma_0(N)\)-optimal
55506.a4 55506d2 \([1, 1, 0, 96698, 13488820]\) \(168105213359/228637728\) \(-135999052674854688\) \([2]\) \(896000\) \(1.9734\)  

Rank

sage: E.rank()
 

The elliptic curves in class 55506.a have rank \(1\).

Complex multiplication

The elliptic curves in class 55506.a do not have complex multiplication.

Modular form 55506.2.a.a

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

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.