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Properties

Label 8670.k
Number of curves $2$
Conductor $8670$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 8670.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8670.k1 8670l2 \([1, 0, 1, -13349928, 18771756406]\) \(10901014250685308569/1040774054400\) \(25121755551489753600\) \([2]\) \(580608\) \(2.7592\)  
8670.k2 8670l1 \([1, 0, 1, -772648, 338494838]\) \(-2113364608155289/828431400960\) \(-19996320102438666240\) \([2]\) \(290304\) \(2.4126\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8670.k have rank \(1\).

Complex multiplication

The elliptic curves in class 8670.k do not have complex multiplication.

Modular form 8670.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2 q^{7} - q^{8} + q^{9} - q^{10} + 4 q^{11} + q^{12} + 4 q^{13} + 2 q^{14} + q^{15} + q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.