↑

Properties

Label 45612.10387
Modulus $45612$
Conductor $5068$
Order $90$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45612, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([45,0,45,86]))
 
Copy content gp:[g,chi] = znchar(Mod(10387, 45612))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45612.10387");
 

Basic properties

Modulus: \(45612\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5068\)
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: no, induced from \(\chi_{5068}(251,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 45612.bla

\(\chi_{45612}(307,\cdot)\) \(\chi_{45612}(559,\cdot)\) \(\chi_{45612}(811,\cdot)\) \(\chi_{45612}(1819,\cdot)\) \(\chi_{45612}(3583,\cdot)\) \(\chi_{45612}(3835,\cdot)\) \(\chi_{45612}(5599,\cdot)\) \(\chi_{45612}(7615,\cdot)\) \(\chi_{45612}(8371,\cdot)\) \(\chi_{45612}(9379,\cdot)\) \(\chi_{45612}(9631,\cdot)\) \(\chi_{45612}(10387,\cdot)\) \(\chi_{45612}(11143,\cdot)\) \(\chi_{45612}(14923,\cdot)\) \(\chi_{45612}(15931,\cdot)\) \(\chi_{45612}(17191,\cdot)\) \(\chi_{45612}(18451,\cdot)\) \(\chi_{45612}(20467,\cdot)\) \(\chi_{45612}(27271,\cdot)\) \(\chi_{45612}(29035,\cdot)\) \(\chi_{45612}(33319,\cdot)\) \(\chi_{45612}(38863,\cdot)\) \(\chi_{45612}(39619,\cdot)\) \(\chi_{45612}(41131,\cdot)\)

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

\((22807,5069,32581,12853)\) → \((-1,1,-1,e\left(\frac{43}{45}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 45612 }(10387, a) \) \(1\)\(1\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{67}{90}\right)\)\(e\left(\frac{19}{90}\right)\)\(e\left(\frac{13}{18}\right)\)\(1\)\(e\left(\frac{13}{90}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{38}{45}\right)\)
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)
 
\( \chi_{ 45612 }(10387,a) \;\) at \(\;a = \) e.g. 2