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Properties

Label 5280.r
Number of curves $4$
Conductor $5280$
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, -8875520, -10180403400]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -8875520, -10180403400]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -8875520, -10180403400]) 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 5280.r 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 \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 5280.r do not have complex multiplication.

Modular form 5280.2.a.r

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

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

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 5280.r

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
5280.r1 5280o2 \([0, 1, 0, -8875520, -10180403400]\) \(151020262560470148771848/35809491031875\) \(18334459408320000\) \([2]\) \(122880\) \(2.4994\)  
5280.r2 5280o3 \([0, 1, 0, -1095825, 197048223]\) \(35529391776305786176/16450653076171875\) \(67381875000000000000\) \([4]\) \(122880\) \(2.4994\)  
5280.r3 5280o1 \([0, 1, 0, -556770, -157973400]\) \(298244193811346574784/4540317078515625\) \(290580293025000000\) \([2, 2]\) \(61440\) \(2.1528\) \(\Gamma_0(N)\)-optimal
5280.r4 5280o4 \([0, 1, 0, -50520, -433980900]\) \(-27851742625371848/158882936571500625\) \(-81348063524608320000\) \([2]\) \(122880\) \(2.4994\)