↑

Properties

Label 70560dk
Number of curves $4$
Conductor $70560$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -8937453, 10256192548]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -8937453, 10256192548]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -8937453, 10256192548]) 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 70560dk have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 70560dk do not have complex multiplication.

Modular form 70560.2.a.dk

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);
 
\(q - q^{5} - 4 q^{11} + 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

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

Elliptic curves in class 70560dk

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
70560.j3 70560dk1 \([0, 0, 0, -8937453, 10256192548]\) \(14383655824793536/45209390625\) \(248155780267521000000\) \([2, 2]\) \(2949120\) \(2.7806\) \(\Gamma_0(N)\)-optimal
70560.j4 70560dk2 \([0, 0, 0, -5186748, 18930823072]\) \(-43927191786304/415283203125\) \(-145888171821000000000000\) \([2]\) \(5898240\) \(3.1271\)  
70560.j2 70560dk3 \([0, 0, 0, -12796203, 537544798]\) \(5276930158229192/3050936350875\) \(133973491831011177408000\) \([2]\) \(5898240\) \(3.1271\)  
70560.j1 70560dk4 \([0, 0, 0, -142891203, 657440340298]\) \(7347751505995469192/72930375\) \(3202537147814592000\) \([2]\) \(5898240\) \(3.1271\)