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Properties

Label 283920ek
Number of curves $6$
Conductor $283920$
CM no
Rank $1$
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, -1014056, -393117516]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -1014056, -393117516]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -1014056, -393117516]) 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 283920ek have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 283920ek do not have complex multiplication.

Modular form 283920.2.a.ek

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

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

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 283920ek

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
283920.ek5 283920ek1 \([0, 1, 0, -1014056, -393117516]\) \(5832972054001/4542720\) \(89812343932846080\) \([2]\) \(4128768\) \(2.1843\) \(\Gamma_0(N)\)-optimal
283920.ek4 283920ek2 \([0, 1, 0, -1230376, -213398860]\) \(10418796526321/5038160400\) \(99607502693022105600\) \([2, 2]\) \(8257536\) \(2.5309\)  
283920.ek2 283920ek3 \([0, 1, 0, -10369896, 12702570804]\) \(6237734630203441/82168222500\) \(1624515853832202240000\) \([2, 2]\) \(16515072\) \(2.8775\)  
283920.ek6 283920ek4 \([0, 1, 0, 4448024, -1623913420]\) \(492271755328079/342606902820\) \(-6773547343846200852480\) \([2]\) \(16515072\) \(2.8775\)  
283920.ek1 283920ek5 \([0, 1, 0, -165390216, 818622210420]\) \(25306558948218234961/4478906250\) \(88550707190400000000\) \([2]\) \(33030144\) \(3.2241\)  
283920.ek3 283920ek6 \([0, 1, 0, -1581896, 33551222004]\) \(-22143063655441/24584858584650\) \(-486057642721754652057600\) \([2]\) \(33030144\) \(3.2241\)