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Properties

Label 209814.cy
Number of curves $4$
Conductor $209814$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cy1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 209814.cy have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(11\)\(1\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 209814.cy do not have complex multiplication.

Modular form 209814.2.a.cy

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

Isogeny matrix

Copy content 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

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 209814.cy

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
209814.cy1 209814g4 \([1, 0, 0, -16922881104, -847326606295212]\) \(12534210458299016895673/315581882565708\) \(13494652383421792645350732972\) \([2]\) \(530841600\) \(4.5058\)  
209814.cy2 209814g2 \([1, 0, 0, -1098009844, -12175355497216]\) \(3423676911662954233/483711578981136\) \(20684075901687405523301057424\) \([2, 2]\) \(265420800\) \(4.1592\)  
209814.cy3 209814g1 \([1, 0, 0, -289526564, 1706140723728]\) \(62768149033310713/6915442583808\) \(295712456581123642732015872\) \([2]\) \(132710400\) \(3.8127\) \(\Gamma_0(N)\)-optimal
209814.cy4 209814g3 \([1, 0, 0, 1791128936, -65436628906516]\) \(14861225463775641287/51859390496937804\) \(-2217568517820699698394138501036\) \([2]\) \(530841600\) \(4.5058\)