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Properties

Label 338910cn
Number of curves $4$
Conductor $338910$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -2469093165, 62319962497617]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -2469093165, 62319962497617]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -2469093165, 62319962497617]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 338910cn have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(11\)\(1 - T\)
\(13\)\(1 - T\)
\(79\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 338910cn do not have complex multiplication.

Modular form 338910.2.a.cn

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 2 q^{7} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} - 2 q^{14} + q^{15} + q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 338910cn

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
338910.cn4 338910cn1 \([1, 0, 0, -2469093165, 62319962497617]\) \(-1664700629958202478932584096762961/714434636321013104640000000000\) \(-714434636321013104640000000000\) \([10]\) \(541440000\) \(4.4360\) \(\Gamma_0(N)\)-optimal
338910.cn2 338910cn2 \([1, 0, 0, -42917093165, 3421777140097617]\) \(8742076626521910595251756451488762961/964148342166520646557593600000\) \(964148342166520646557593600000\) \([10]\) \(1082880000\) \(4.7826\)  
338910.cn3 338910cn3 \([1, 0, 0, -19953261165, -6053922020220783]\) \(-878547754640224469469830258471034961/15324360292462134906363408448382400\) \(-15324360292462134906363408448382400\) \([2]\) \(2707200000\) \(5.2407\)  
338910.cn1 338910cn4 \([1, 0, 0, -635364540965, -194182565527705623]\) \(28365643022300454160291628196841004894161/125944255938608020119607441810787160\) \(125944255938608020119607441810787160\) \([2]\) \(5414400000\) \(5.5873\)