Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -4920320, 4202499840]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -4920320, 4202499840]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -4920320, 4202499840]) 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 7440.j have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(31\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 7440.j do not have complex multiplication.

Modular form 7440.2.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);
 
\(q - q^{3} + q^{5} + q^{9} + 4 q^{11} + 6 q^{13} - q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 7440.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
7440.j1 7440o5 \([0, -1, 0, -4920320, 4202499840]\) \(3216206300355197383681/57660\) \(236175360\) \([4]\) \(98304\) \(2.0750\)  
7440.j2 7440o4 \([0, -1, 0, -307520, 65740800]\) \(785209010066844481/3324675600\) \(13617871257600\) \([2, 4]\) \(49152\) \(1.7284\)  
7440.j3 7440o6 \([0, -1, 0, -302720, 67887360]\) \(-749011598724977281/51173462246460\) \(-209606501361500160\) \([4]\) \(98304\) \(2.0750\)  
7440.j4 7440o3 \([0, -1, 0, -59200, -4302848]\) \(5601911201812801/1271193750000\) \(5206809600000000\) \([2]\) \(49152\) \(1.7284\)  
7440.j5 7440o2 \([0, -1, 0, -19520, 998400]\) \(200828550012481/12454560000\) \(51013877760000\) \([2, 2]\) \(24576\) \(1.3818\)  
7440.j6 7440o1 \([0, -1, 0, 960, 64512]\) \(23862997439/457113600\) \(-1872337305600\) \([2]\) \(12288\) \(1.0353\) \(\Gamma_0(N)\)-optimal