Properties

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

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Show commands: SageMath
E = EllipticCurve("fw1")
 
E.isogeny_class()
 

Elliptic curves in class 163170fw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
163170.o2 163170fw1 \([1, -1, 0, -16620, 1002896]\) \(-219256227/59200\) \(-137088567806400\) \([2]\) \(663552\) \(1.4289\) \(\Gamma_0(N)\)-optimal
163170.o1 163170fw2 \([1, -1, 0, -281220, 57468536]\) \(1062144635427/54760\) \(126806925220920\) \([2]\) \(1327104\) \(1.7755\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 163170.2.a.fw

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 Cremona 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 Cremona labels.