Properties

Label 324870.ck
Number of curves $4$
Conductor $324870$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ck1")
 
E.isogeny_class()
 

Elliptic curves in class 324870.ck

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
324870.ck1 324870ck3 \([1, 0, 1, -2013583, 966826298]\) \(7674388308884766169/1007648705929320\) \(118548862603878568680\) \([2]\) \(14155776\) \(2.5800\)  
324870.ck2 324870ck2 \([1, 0, 1, -514183, -126536182]\) \(127787213284071769/15197834433600\) \(1788010023278606400\) \([2, 2]\) \(7077888\) \(2.2334\)  
324870.ck3 324870ck1 \([1, 0, 1, -498503, -135511414]\) \(116449478628435289/1996001280\) \(234827554590720\) \([2]\) \(3538944\) \(1.8868\) \(\Gamma_0(N)\)-optimal
324870.ck4 324870ck4 \([1, 0, 1, 734337, -645421094]\) \(372239584720800551/1745320379985000\) \(-205335197384855265000\) \([2]\) \(14155776\) \(2.5800\)  

Rank

sage: E.rank()
 

The elliptic curves in class 324870.ck have rank \(1\).

Complex multiplication

The elliptic curves in class 324870.ck do not have complex multiplication.

Modular form 324870.2.a.ck

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} + q^{13} + q^{15} + q^{16} - q^{17} - q^{18} + O(q^{20})\) 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.