Properties

Label 55a
Number of curves $4$
Conductor $55$
CM no
Rank $0$
Graph

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

Elliptic curves in class 55a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
55.a3 55a1 \([1, -1, 0, -4, 3]\) \(8120601/3025\) \(3025\) \([2, 2]\) \(2\) \(-0.63224\) \(\Gamma_0(N)\)-optimal
55.a2 55a2 \([1, -1, 0, -29, -52]\) \(2749884201/73205\) \(73205\) \([2]\) \(4\) \(-0.28567\)  
55.a1 55a3 \([1, -1, 0, -59, 190]\) \(22930509321/6875\) \(6875\) \([4]\) \(4\) \(-0.28567\)  
55.a4 55a4 \([1, -1, 0, 1, 0]\) \(59319/55\) \(-55\) \([2]\) \(4\) \(-0.97881\)  

Rank

sage: E.rank()
 

The elliptic curves in class 55a have rank \(0\).

Complex multiplication

The elliptic curves in class 55a do not have complex multiplication.

Modular form 55.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} + q^{5} - 3q^{8} - 3q^{9} + q^{10} - q^{11} + 2q^{13} - q^{16} + 6q^{17} - 3q^{18} - 4q^{19} + O(q^{20})\)  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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.