Properties

Label 163170.c
Number of curves $4$
Conductor $163170$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 163170.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
163170.c1 163170ed4 \([1, -1, 0, -5221890, -4591623240]\) \(183607808836587409/5330220\) \(457152293476620\) \([2]\) \(5111808\) \(2.3215\)  
163170.c2 163170ed3 \([1, -1, 0, -512010, 18862416]\) \(173078750185489/98393452500\) \(8438824752722752500\) \([2]\) \(5111808\) \(2.3215\)  
163170.c3 163170ed2 \([1, -1, 0, -326790, -71487900]\) \(45000254125009/241491600\) \(20711797786083600\) \([2, 2]\) \(2555904\) \(1.9749\)  
163170.c4 163170ed1 \([1, -1, 0, -9270, -2332044]\) \(-1027243729/26853120\) \(-2303087939147520\) \([2]\) \(1277952\) \(1.6283\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 163170.c have rank \(2\).

Complex multiplication

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

Modular form 163170.2.a.c

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.