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Properties

Label 6336bg
Number of curves $4$
Conductor $6336$
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, -25932, -1171600]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -25932, -1171600]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -25932, -1171600]) 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 6336bg have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 6336bg do not have complex multiplication.

Modular form 6336.2.a.bg

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);
 
\(q - 4 q^{5} - 2 q^{7} + q^{11} - 4 q^{13} + 2 q^{17} + O(q^{20})\) 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 6336bg

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
6336.c3 6336bg1 \([0, 0, 0, -25932, -1171600]\) \(10091699281/2737152\) \(523077892964352\) \([2]\) \(30720\) \(1.5322\) \(\Gamma_0(N)\)-optimal
6336.c4 6336bg2 \([0, 0, 0, 66228, -7622800]\) \(168105213359/228637728\) \(-43693350246678528\) \([2]\) \(61440\) \(1.8787\)  
6336.c1 6336bg3 \([0, 0, 0, -5797452, 5372839280]\) \(112763292123580561/1932612\) \(369327904653312\) \([2]\) \(153600\) \(2.3369\)  
6336.c2 6336bg4 \([0, 0, 0, -5791692, 5384048240]\) \(-112427521449300721/466873642818\) \(-89220942558480826368\) \([2]\) \(307200\) \(2.6835\)