Properties

Label 171.a
Number of curves $4$
Conductor $171$
CM no
Rank $0$
Graph

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

Elliptic curves in class 171.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
171.a1 171a3 \([1, -1, 1, -914, -10402]\) \(115714886617/1539\) \(1121931\) \([2]\) \(48\) \(0.30430\)  
171.a2 171a2 \([1, -1, 1, -59, -142]\) \(30664297/3249\) \(2368521\) \([2, 2]\) \(24\) \(-0.042276\)  
171.a3 171a1 \([1, -1, 1, -14, 20]\) \(389017/57\) \(41553\) \([4]\) \(12\) \(-0.38885\) \(\Gamma_0(N)\)-optimal
171.a4 171a4 \([1, -1, 1, 76, -790]\) \(67419143/390963\) \(-285012027\) \([2]\) \(48\) \(0.30430\)  

Rank

sage: E.rank()
 

The elliptic curves in class 171.a have rank \(0\).

Complex multiplication

The elliptic curves in class 171.a do not have complex multiplication.

Modular form 171.2.a.a

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