↑

Properties

Label 6384.2123
Modulus $6384$
Conductor $6384$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6384, base_ring=CyclotomicField(36)) M = H._module chi = DirichletCharacter(H, M([18,9,18,12,14]))
 
Copy content gp:[g,chi] = znchar(Mod(2123, 6384))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6384.2123");
 

Basic properties

Modulus: \(6384\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6384\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(36\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 6384.nh

\(\chi_{6384}(611,\cdot)\) \(\chi_{6384}(851,\cdot)\) \(\chi_{6384}(2027,\cdot)\) \(\chi_{6384}(2123,\cdot)\) \(\chi_{6384}(2195,\cdot)\) \(\chi_{6384}(2795,\cdot)\) \(\chi_{6384}(3803,\cdot)\) \(\chi_{6384}(4043,\cdot)\) \(\chi_{6384}(5219,\cdot)\) \(\chi_{6384}(5315,\cdot)\) \(\chi_{6384}(5387,\cdot)\) \(\chi_{6384}(5987,\cdot)\)

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((799,4789,2129,913,1009)\) → \((-1,i,-1,e\left(\frac{1}{3}\right),e\left(\frac{7}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 6384 }(2123, a) \) \(-1\)\(1\)\(e\left(\frac{23}{36}\right)\)\(i\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{18}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 6384 }(2123,a) \;\) at \(\;a = \) e.g. 2