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Properties

Label 6479.4551
Modulus $6479$
Conductor $6479$
Order $90$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6479, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([27,85,30]))
 
Copy content gp:[g,chi] = znchar(Mod(4551, 6479))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6479.4551");
 

Basic properties

Modulus: \(6479\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6479\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(90\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 6479.ql

\(\chi_{6479}(470,\cdot)\) \(\chi_{6479}(490,\cdot)\) \(\chi_{6479}(1172,\cdot)\) \(\chi_{6479}(1668,\cdot)\) \(\chi_{6479}(1865,\cdot)\) \(\chi_{6479}(2195,\cdot)\) \(\chi_{6479}(2206,\cdot)\) \(\chi_{6479}(2257,\cdot)\) \(\chi_{6479}(2350,\cdot)\) \(\chi_{6479}(2846,\cdot)\) \(\chi_{6479}(2939,\cdot)\) \(\chi_{6479}(3043,\cdot)\) \(\chi_{6479}(3373,\cdot)\) \(\chi_{6479}(3384,\cdot)\) \(\chi_{6479}(3528,\cdot)\) \(\chi_{6479}(3632,\cdot)\) \(\chi_{6479}(3962,\cdot)\) \(\chi_{6479}(3973,\cdot)\) \(\chi_{6479}(4221,\cdot)\) \(\chi_{6479}(4551,\cdot)\) \(\chi_{6479}(4562,\cdot)\) \(\chi_{6479}(4593,\cdot)\) \(\chi_{6479}(5771,\cdot)\) \(\chi_{6479}(6360,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((1179,6139,4808)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{17}{18}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 6479 }(4551, a) \) \(1\)\(1\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{1}{90}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{23}{90}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{2}{9}\right)\)\(-1\)
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)
 
\( \chi_{ 6479 }(4551,a) \;\) at \(\;a = \) e.g. 2