↑

Properties

Label 66c
Number of curves $4$
Conductor $66$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -45, 81]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -45, 81]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -45, 81]) 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 66c have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 4 T + 5 T^{2}\) 1.5.e
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 66c do not have complex multiplication.

Modular form 66.2.a.c

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

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 66c

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
66.c3 66c1 \([1, 0, 0, -45, 81]\) \(10091699281/2737152\) \(2737152\) \([10]\) \(20\) \(-0.056866\) \(\Gamma_0(N)\)-optimal
66.c4 66c2 \([1, 0, 0, 115, 561]\) \(168105213359/228637728\) \(-228637728\) \([10]\) \(40\) \(0.28971\)  
66.c1 66c3 \([1, 0, 0, -10065, -389499]\) \(112763292123580561/1932612\) \(1932612\) \([2]\) \(100\) \(0.74785\)  
66.c2 66c4 \([1, 0, 0, -10055, -390309]\) \(-112427521449300721/466873642818\) \(-466873642818\) \([2]\) \(200\) \(1.0944\)