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Properties

Label 12551.139
Modulus $12551$
Conductor $12551$
Order $810$
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(12551, base_ring=CyclotomicField(810)) M = H._module chi = DirichletCharacter(H, M([405,567,115]))
 
Copy content gp:[g,chi] = znchar(Mod(139, 12551))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12551.139");
 

Basic properties

Modulus: \(12551\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(12551\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(810\)
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 12551.il

\(\chi_{12551}(139,\cdot)\) \(\chi_{12551}(195,\cdot)\) \(\chi_{12551}(272,\cdot)\) \(\chi_{12551}(293,\cdot)\) \(\chi_{12551}(370,\cdot)\) \(\chi_{12551}(398,\cdot)\) \(\chi_{12551}(475,\cdot)\) \(\chi_{12551}(552,\cdot)\) \(\chi_{12551}(601,\cdot)\) \(\chi_{12551}(734,\cdot)\) \(\chi_{12551}(755,\cdot)\) \(\chi_{12551}(776,\cdot)\) \(\chi_{12551}(811,\cdot)\) \(\chi_{12551}(860,\cdot)\) \(\chi_{12551}(888,\cdot)\) \(\chi_{12551}(909,\cdot)\) \(\chi_{12551}(937,\cdot)\) \(\chi_{12551}(1007,\cdot)\) \(\chi_{12551}(1084,\cdot)\) \(\chi_{12551}(1161,\cdot)\) \(\chi_{12551}(1217,\cdot)\) \(\chi_{12551}(1294,\cdot)\) \(\chi_{12551}(1315,\cdot)\) \(\chi_{12551}(1322,\cdot)\) \(\chi_{12551}(1371,\cdot)\) \(\chi_{12551}(1469,\cdot)\) \(\chi_{12551}(1546,\cdot)\)

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: $\Q(\zeta_{405})$
Fixed field: Number field defined by a degree 810 polynomial (not computed)

Values on generators

\((3587,7988,2773)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{23}{162}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 12551 }(139, a) \) \(-1\)\(1\)\(e\left(\frac{341}{405}\right)\)\(e\left(\frac{178}{405}\right)\)\(e\left(\frac{277}{405}\right)\)\(e\left(\frac{58}{135}\right)\)\(e\left(\frac{38}{135}\right)\)\(e\left(\frac{71}{135}\right)\)\(e\left(\frac{356}{405}\right)\)\(e\left(\frac{22}{81}\right)\)\(e\left(\frac{10}{81}\right)\)\(e\left(\frac{119}{270}\right)\)
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)
 
\( \chi_{ 12551 }(139,a) \;\) at \(\;a = \) e.g. 2