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Properties

Label 2640.j
Number of curves $4$
Conductor $2640$
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([0, 1, 0, -29326496, 61113921780]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -29326496, 61113921780]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -29326496, 61113921780]) 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 2640.j have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 2640.j do not have complex multiplication.

Modular form 2640.2.a.j

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);
 
\(q + q^{3} - q^{5} - 4 q^{7} + q^{9} + q^{11} + 2 q^{13} - q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

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 2640.j

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
2640.j1 2640u3 \([0, 1, 0, -29326496, 61113921780]\) \(680995599504466943307169/52207031250000000\) \(213840000000000000000\) \([2]\) \(215040\) \(2.9492\)  
2640.j2 2640u2 \([0, 1, 0, -1954976, 819937524]\) \(201738262891771037089/45727545600000000\) \(187300026777600000000\) \([2, 2]\) \(107520\) \(2.6026\)  
2640.j3 2640u1 \([0, 1, 0, -644256, -188268300]\) \(7220044159551112609/448454983680000\) \(1836871613153280000\) \([2]\) \(53760\) \(2.2561\) \(\Gamma_0(N)\)-optimal
2640.j4 2640u4 \([0, 1, 0, 4445024, 5072097524]\) \(2371297246710590562911/4084000833203280000\) \(-16728067412800634880000\) \([4]\) \(215040\) \(2.9492\)