Properties

Label 161.a
Number of curves $4$
Conductor $161$
CM no
Rank $0$
Graph

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

Elliptic curves in class 161.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
161.a1 161a3 \([1, -1, 1, -124, 560]\) \(209267191953/55223\) \(55223\) \([4]\) \(20\) \(-0.10563\)  
161.a2 161a1 \([1, -1, 1, -9, 8]\) \(72511713/25921\) \(25921\) \([2, 2]\) \(10\) \(-0.45221\) \(\Gamma_0(N)\)-optimal
161.a3 161a2 \([1, -1, 1, -4, -2]\) \(5545233/161\) \(161\) \([2]\) \(20\) \(-0.79878\)  
161.a4 161a4 \([1, -1, 1, 26, 36]\) \(2014698447/1958887\) \(-1958887\) \([2]\) \(20\) \(-0.10563\)  

Rank

sage: E.rank()
 

The elliptic curves in class 161.a have rank \(0\).

Complex multiplication

The elliptic curves in class 161.a do not have complex multiplication.

Modular form 161.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + 2q^{5} + q^{7} + 3q^{8} - 3q^{9} - 2q^{10} + 4q^{11} + 6q^{13} - q^{14} - q^{16} - 2q^{17} + 3q^{18} + 4q^{19} + O(q^{20})\)  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 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.