Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, -1429, -20893]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -1429, -20893]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -1429, -20893]) 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 795.d have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(53\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 795.d do not have complex multiplication.

Modular form 795.2.a.d

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);
 
\(q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - 3 q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + 6 q^{13} - q^{15} - q^{16} + 6 q^{17} + q^{18} + O(q^{20})\) 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 labeled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 795.d

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
795.d1 795d3 \([1, 0, 1, -1429, -20893]\) \(322391399464009/140831865\) \(140831865\) \([2]\) \(480\) \(0.52294\)  
795.d2 795d4 \([1, 0, 1, -779, 8147]\) \(52183647114409/1065214935\) \(1065214935\) \([2]\) \(480\) \(0.52294\)  
795.d3 795d2 \([1, 0, 1, -104, -223]\) \(122689385209/51194025\) \(51194025\) \([2, 2]\) \(240\) \(0.17637\)  
795.d4 795d1 \([1, 0, 1, 21, -23]\) \(1095912791/894375\) \(-894375\) \([2]\) \(120\) \(-0.17021\) \(\Gamma_0(N)\)-optimal