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Properties

Label 277440.gf
Number of curves $4$
Conductor $277440$
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, -3440641, -2452455841]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -3440641, -2452455841]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -3440641, -2452455841]) 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 277440.gf have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(17\)\(1\)
 
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
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 277440.gf do not have complex multiplication.

Modular form 277440.2.a.gf

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

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

Isogeny graph

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

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

Elliptic curves in class 277440.gf

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
277440.gf1 277440gf4 \([0, 1, 0, -3440641, -2452455841]\) \(711882749089/1721250\) \(10891241885859840000\) \([2]\) \(7077888\) \(2.5317\)  
277440.gf2 277440gf3 \([0, 1, 0, -3070721, 2061234015]\) \(506071034209/2505630\) \(15854421151179079680\) \([2]\) \(7077888\) \(2.5317\)  
277440.gf3 277440gf2 \([0, 1, 0, -296321, -6803745]\) \(454756609/260100\) \(1645787662752153600\) \([2, 2]\) \(3538944\) \(2.1851\)  
277440.gf4 277440gf1 \([0, 1, 0, 73599, -811041]\) \(6967871/4080\) \(-25816277062778880\) \([2]\) \(1769472\) \(1.8386\) \(\Gamma_0(N)\)-optimal