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Properties

Label 159120.ek
Number of curves $8$
Conductor $159120$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -949104000867, -355892486813447326]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -949104000867, -355892486813447326]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -949104000867, -355892486813447326]) 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 159120.ek have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(13\)\(1 - T\)
\(17\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 159120.ek do not have complex multiplication.

Modular form 159120.2.a.ek

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);
 
\(q + q^{5} + 4 q^{7} + q^{13} - q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 159120.ek

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
159120.ek1 159120z8 \([0, 0, 0, -949104000867, -355892486813447326]\) \(31664865542564944883878115208137569/103216295812500\) \(308202207835392000000\) \([2]\) \(637009920\) \(5.0166\)  
159120.ek2 159120z6 \([0, 0, 0, -59319000867, -5560819946447326]\) \(7730680381889320597382223137569/441370202660156250000\) \(1317924363219984000000000000\) \([2, 2]\) \(318504960\) \(4.6701\)  
159120.ek3 159120z7 \([0, 0, 0, -59211435747, -5581991709347614]\) \(-7688701694683937879808871873249/58423707246780395507812500\) \(-174452255059570312500000000000000\) \([2]\) \(637009920\) \(5.0166\)  
159120.ek4 159120z5 \([0, 0, 0, -11717750307, -488156229494494]\) \(59589391972023341137821784609/8834417507562311995200\) \(26379429326900942620675276800\) \([2]\) \(212336640\) \(4.4673\)  
159120.ek5 159120z3 \([0, 0, 0, -3714161187, -86556842855134]\) \(1897660325010178513043539489/14258428094958372000000\) \(42575438156696179458048000000\) \([2]\) \(159252480\) \(4.3235\)  
159120.ek6 159120z2 \([0, 0, 0, -800246307, -6128776387294]\) \(18980483520595353274840609/5549773448629762560000\) \(16571534721233292927959040000\) \([2, 2]\) \(106168320\) \(4.1208\)  
159120.ek7 159120z1 \([0, 0, 0, -301845027, 1943629784354]\) \(1018563973439611524445729/42904970360310988800\) \(128113555016362847580979200\) \([2]\) \(53084160\) \(3.7742\) \(\Gamma_0(N)\)-optimal
159120.ek8 159120z4 \([0, 0, 0, 2142837213, -40735318265566]\) \(364421318680576777174674911/450962301637624725000000\) \(-1346566217293121226854400000000\) \([2]\) \(212336640\) \(4.4673\)