Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, 1340, -4051]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, 1340, -4051]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, 1340, -4051]) 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 28749e have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(7\)\(1 - T\)
\(37\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + 2 T^{2}\) 1.2.a
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(11\) \( 1 + T + 11 T^{2}\) 1.11.b
\(13\) \( 1 - 5 T + 13 T^{2}\) 1.13.af
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 8 T + 29 T^{2}\) 1.29.i
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 28749e do not have complex multiplication.

Modular form 28749.2.a.e

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);
 
\(q + q^{2} + q^{3} - q^{4} + 2 q^{5} + q^{6} - q^{7} - 3 q^{8} + q^{9} + 2 q^{10} + 4 q^{11} - q^{12} + 2 q^{13} - q^{14} + 2 q^{15} - q^{16} + 6 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

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 28749e

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
28749.h6 28749e1 \([1, 0, 1, 1340, -4051]\) \(103823/63\) \(-161640763767\) \([2]\) \(25344\) \(0.83994\) \(\Gamma_0(N)\)-optimal
28749.h5 28749e2 \([1, 0, 1, -5505, -34169]\) \(7189057/3969\) \(10183368117321\) \([2, 2]\) \(50688\) \(1.1865\)  
28749.h3 28749e3 \([1, 0, 1, -53420, 4718999]\) \(6570725617/45927\) \(117836116786143\) \([2]\) \(101376\) \(1.5331\)  
28749.h2 28749e4 \([1, 0, 1, -67110, -6687509]\) \(13027640977/21609\) \(55442781972081\) \([2, 2]\) \(101376\) \(1.5331\)  
28749.h4 28749e5 \([1, 0, 1, -46575, -10852007]\) \(-4354703137/17294403\) \(-44372706504988827\) \([2]\) \(202752\) \(1.8797\)  
28749.h1 28749e6 \([1, 0, 1, -1073325, -428090351]\) \(53297461115137/147\) \(377161782123\) \([2]\) \(202752\) \(1.8797\)