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Properties

Label 465690.k
Number of curves $6$
Conductor $465690$
CM no
Rank $1$
Graph

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

Elliptic curves in class 465690.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
465690.k1 465690k5 \([1, 1, 0, -405838012, 3146689270486]\) \(157130420902139847946321/82338281250\) \(3873676981432031250\) \([2]\) \(82575360\) \(3.3332\) \(\Gamma_0(N)\)-optimal*
465690.k2 465690k3 \([1, 1, 0, -25369282, 49141152064]\) \(38381916934612839601/27769211122500\) \(1306427001933011422500\) \([2, 2]\) \(41287680\) \(2.9866\) \(\Gamma_0(N)\)-optimal*
465690.k3 465690k6 \([1, 1, 0, -20225032, 69657449914]\) \(-19447769219685987601/33311370791162850\) \(-1567162786187923292720850\) \([2]\) \(82575360\) \(3.3332\)  
465690.k4 465690k4 \([1, 1, 0, -15882202, -24073087544]\) \(9417471079857004081/131452777937340\) \(6184311747959523096540\) \([2]\) \(41287680\) \(2.9866\)  
465690.k5 465690k2 \([1, 1, 0, -1911502, 428726116]\) \(16418244983975281/7807218339600\) \(367297464945839187600\) \([2, 2]\) \(20643840\) \(2.6401\) \(\Gamma_0(N)\)-optimal*
465690.k6 465690k1 \([1, 1, 0, 427778, 51166324]\) \(184016114839439/130363395840\) \(-6133060807444535040\) \([2]\) \(10321920\) \(2.2935\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 465690.k1.

Rank

sage: E.rank()
 

The elliptic curves in class 465690.k have rank \(1\).

Complex multiplication

The elliptic curves in class 465690.k do not have complex multiplication.

Modular form 465690.2.a.k

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.