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Properties

Label 136290a
Number of curves $4$
Conductor $136290$
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, 31710490, 2664272100]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, 31710490, 2664272100]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, 31710490, 2664272100]) 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 136290a have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(7\)\(1 + T\)
\(11\)\(1 + T\)
\(59\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 136290a do not have complex multiplication.

Modular form 136290.2.a.a

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);
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} - 6 q^{13} + q^{14} + q^{15} + q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) 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 136290a

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
136290.ct4 136290a1 \([1, 0, 0, 31710490, 2664272100]\) \(3526408097252790554911347359/2043805832116142371200000\) \(-2043805832116142371200000\) \([10]\) \(32320000\) \(3.3542\) \(\Gamma_0(N)\)-optimal
136290.ct3 136290a2 \([1, 0, 0, -126947590, 21290730692]\) \(226254431386145697240615608161/130738259239901991562500000\) \(130738259239901991562500000\) \([10]\) \(64640000\) \(3.7008\)  
136290.ct2 136290a3 \([1, 0, 0, -3336162410, -74180003513280]\) \(-4106437962229085333075261665806241/726025922559946496763017820\) \(-726025922559946496763017820\) \([2]\) \(161600000\) \(4.1589\)  
136290.ct1 136290a4 \([1, 0, 0, -53380863340, -4747083934971658]\) \(16822110927907071112393084202410196161/525702416105845179466650\) \(525702416105845179466650\) \([2]\) \(323200000\) \(4.5055\)