Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -579, 11842]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -579, 11842]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -579, 11842]) 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 1008l have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1008l do not have complex multiplication.

Modular form 1008.2.a.l

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);
 
\(q + 2 q^{5} + q^{7} - 4 q^{11} + 6 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

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

Elliptic curves in class 1008l

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
1008.j5 1008l1 \([0, 0, 0, -579, 11842]\) \(-7189057/16128\) \(-48157949952\) \([2]\) \(768\) \(0.73841\) \(\Gamma_0(N)\)-optimal
1008.j4 1008l2 \([0, 0, 0, -12099, 511810]\) \(65597103937/63504\) \(189621927936\) \([2, 2]\) \(1536\) \(1.0850\)  
1008.j3 1008l3 \([0, 0, 0, -14979, 249730]\) \(124475734657/63011844\) \(188152357994496\) \([2, 2]\) \(3072\) \(1.4316\)  
1008.j1 1008l4 \([0, 0, 0, -193539, 32771842]\) \(268498407453697/252\) \(752467968\) \([2]\) \(3072\) \(1.4316\)  
1008.j2 1008l5 \([0, 0, 0, -131619, -18202718]\) \(84448510979617/933897762\) \(2788603774967808\) \([2]\) \(6144\) \(1.7781\)  
1008.j6 1008l6 \([0, 0, 0, 55581, 1929058]\) \(6359387729183/4218578658\) \(-12596608375529472\) \([2]\) \(6144\) \(1.7781\)