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Properties

Label 11243.5
Modulus $11243$
Conductor $11243$
Order $11242$
Real no
Primitive yes
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(11243, base_ring=CyclotomicField(11242)) M = H._module chi = DirichletCharacter(H, M([1]))
 
Copy content gp:[g,chi] = znchar(Mod(5, 11243))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11243.5");
 

Basic properties

Modulus: \(11243\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11243\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(11242\)
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 11243.p

\(\chi_{11243}(5,\cdot)\) \(\chi_{11243}(7,\cdot)\) \(\chi_{11243}(11,\cdot)\) \(\chi_{11243}(13,\cdot)\) \(\chi_{11243}(15,\cdot)\) \(\chi_{11243}(20,\cdot)\) \(\chi_{11243}(21,\cdot)\) \(\chi_{11243}(28,\cdot)\) \(\chi_{11243}(33,\cdot)\) \(\chi_{11243}(37,\cdot)\) \(\chi_{11243}(38,\cdot)\) \(\chi_{11243}(39,\cdot)\) \(\chi_{11243}(44,\cdot)\) \(\chi_{11243}(45,\cdot)\) \(\chi_{11243}(46,\cdot)\) \(\chi_{11243}(50,\cdot)\) \(\chi_{11243}(52,\cdot)\) \(\chi_{11243}(58,\cdot)\) \(\chi_{11243}(60,\cdot)\) \(\chi_{11243}(63,\cdot)\) \(\chi_{11243}(71,\cdot)\) \(\chi_{11243}(79,\cdot)\) \(\chi_{11243}(80,\cdot)\) \(\chi_{11243}(82,\cdot)\) \(\chi_{11243}(83,\cdot)\) \(\chi_{11243}(84,\cdot)\) \(\chi_{11243}(86,\cdot)\) \(\chi_{11243}(89,\cdot)\) \(\chi_{11243}(94,\cdot)\) \(\chi_{11243}(95,\cdot)\) ...

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: $\Q(\zeta_{5621})$
Fixed field: Number field defined by a degree 11242 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{1}{11242}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 11243 }(5, a) \) \(-1\)\(1\)\(e\left(\frac{519}{1606}\right)\)\(e\left(\frac{57}{73}\right)\)\(e\left(\frac{519}{803}\right)\)\(e\left(\frac{1}{11242}\right)\)\(e\left(\frac{167}{1606}\right)\)\(e\left(\frac{9951}{11242}\right)\)\(e\left(\frac{1557}{1606}\right)\)\(e\left(\frac{41}{73}\right)\)\(e\left(\frac{1817}{5621}\right)\)\(e\left(\frac{4883}{11242}\right)\)
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)
 
\( \chi_{ 11243 }(5,a) \;\) at \(\;a = \) e.g. 2