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Properties

Label 12245.1092
Modulus $12245$
Conductor $12245$
Order $780$
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(12245, base_ring=CyclotomicField(780)) M = H._module chi = DirichletCharacter(H, M([195,728,180]))
 
Copy content gp:[g,chi] = znchar(Mod(1092, 12245))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12245.1092");
 

Basic properties

Modulus: \(12245\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(12245\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(780\)
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 12245.is

\(\chi_{12245}(18,\cdot)\) \(\chi_{12245}(38,\cdot)\) \(\chi_{12245}(143,\cdot)\) \(\chi_{12245}(258,\cdot)\) \(\chi_{12245}(338,\cdot)\) \(\chi_{12245}(413,\cdot)\) \(\chi_{12245}(417,\cdot)\) \(\chi_{12245}(462,\cdot)\) \(\chi_{12245}(617,\cdot)\) \(\chi_{12245}(732,\cdot)\) \(\chi_{12245}(733,\cdot)\) \(\chi_{12245}(763,\cdot)\) \(\chi_{12245}(857,\cdot)\) \(\chi_{12245}(877,\cdot)\) \(\chi_{12245}(887,\cdot)\) \(\chi_{12245}(958,\cdot)\) \(\chi_{12245}(1012,\cdot)\) \(\chi_{12245}(1037,\cdot)\) \(\chi_{12245}(1073,\cdot)\) \(\chi_{12245}(1092,\cdot)\) \(\chi_{12245}(1223,\cdot)\) \(\chi_{12245}(1237,\cdot)\) \(\chi_{12245}(1247,\cdot)\) \(\chi_{12245}(1353,\cdot)\) \(\chi_{12245}(1547,\cdot)\)

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: $\Q(\zeta_{780})$
Fixed field: Number field defined by a degree 780 polynomial (not computed)

Values on generators

\((9797,9086,8061)\) → \((i,e\left(\frac{14}{15}\right),e\left(\frac{3}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 12245 }(1092, a) \) \(-1\)\(1\)\(e\left(\frac{149}{260}\right)\)\(e\left(\frac{713}{780}\right)\)\(e\left(\frac{19}{130}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{479}{780}\right)\)\(e\left(\frac{187}{260}\right)\)\(e\left(\frac{323}{390}\right)\)\(e\left(\frac{31}{195}\right)\)\(e\left(\frac{47}{780}\right)\)\(e\left(\frac{673}{780}\right)\)
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)
 
\( \chi_{ 12245 }(1092,a) \;\) at \(\;a = \) e.g. 2