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Properties

Label 5724.3887
Modulus $5724$
Conductor $636$
Order $52$
Real no
Primitive no
Minimal yes
Parity odd

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(5724, base_ring=CyclotomicField(52)) M = H._module chi = DirichletCharacter(H, M([26,26,35]))
 
Copy content gp:[g,chi] = znchar(Mod(3887, 5724))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5724.3887");
 

Basic properties

Modulus: \(5724\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(636\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(52\)
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: no, induced from \(\chi_{636}(71,\cdot)\)
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 5724.bu

\(\chi_{5724}(215,\cdot)\) \(\chi_{5724}(323,\cdot)\) \(\chi_{5724}(1079,\cdot)\) \(\chi_{5724}(1187,\cdot)\) \(\chi_{5724}(1511,\cdot)\) \(\chi_{5724}(1727,\cdot)\) \(\chi_{5724}(1835,\cdot)\) \(\chi_{5724}(1943,\cdot)\) \(\chi_{5724}(2159,\cdot)\) \(\chi_{5724}(2267,\cdot)\) \(\chi_{5724}(2483,\cdot)\) \(\chi_{5724}(2807,\cdot)\) \(\chi_{5724}(3023,\cdot)\) \(\chi_{5724}(3347,\cdot)\) \(\chi_{5724}(3563,\cdot)\) \(\chi_{5724}(3671,\cdot)\) \(\chi_{5724}(3887,\cdot)\) \(\chi_{5724}(3995,\cdot)\) \(\chi_{5724}(4103,\cdot)\) \(\chi_{5724}(4319,\cdot)\) \(\chi_{5724}(4643,\cdot)\) \(\chi_{5724}(4751,\cdot)\) \(\chi_{5724}(5507,\cdot)\) \(\chi_{5724}(5615,\cdot)\)

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: $\Q(\zeta_{52})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 52 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((2863,4241,2917)\) → \((-1,-1,e\left(\frac{35}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 5724 }(3887, a) \) \(-1\)\(1\)\(e\left(\frac{7}{52}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{21}{52}\right)\)\(i\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{37}{52}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 5724 }(3887,a) \;\) at \(\;a = \) e.g. 2