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Properties

Label 163170.gb
Number of curves $2$
Conductor $163170$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("gb1")
 
E.isogeny_class()
 

Elliptic curves in class 163170.gb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
163170.gb1 163170ch2 \([1, -1, 1, -31247, -2118049]\) \(1062144635427/54760\) \(173946399480\) \([2]\) \(442368\) \(1.2262\)  
163170.gb2 163170ch1 \([1, -1, 1, -1847, -36529]\) \(-219256227/59200\) \(-188050161600\) \([2]\) \(221184\) \(0.87959\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 163170.gb have rank \(1\).

Complex multiplication

The elliptic curves in class 163170.gb do not have complex multiplication.

Modular form 163170.2.a.gb

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} + 2 q^{11} + 2 q^{13} + q^{16} + 2 q^{17} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.