Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -8763602, 9981900641]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -8763602, 9981900641]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -8763602, 9981900641]) 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 6762.bc have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(7\)\(1\)
\(23\)\(1 - T\)
 
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.g
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 6762.bc do not have complex multiplication.

Modular form 6762.2.a.bc

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

Elliptic curves in class 6762.bc

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
6762.bc1 6762bb4 \([1, 1, 1, -8763602, 9981900641]\) \(632678989847546725777/80515134\) \(9472524999966\) \([2]\) \(184320\) \(2.3499\)  
6762.bc2 6762bb3 \([1, 1, 1, -626662, 107865953]\) \(231331938231569617/90942310746882\) \(10699271917059920418\) \([2]\) \(184320\) \(2.3499\)  
6762.bc3 6762bb2 \([1, 1, 1, -547772, 155767961]\) \(154502321244119857/55101928644\) \(6482686803037956\) \([2, 2]\) \(92160\) \(2.0034\)  
6762.bc4 6762bb1 \([1, 1, 1, -29352, 3145113]\) \(-23771111713777/22848457968\) \(-2688098231477232\) \([4]\) \(46080\) \(1.6568\) \(\Gamma_0(N)\)-optimal