Properties

Label 1728.721
Modulus $1728$
Conductor $144$
Order $12$
Real no
Primitive no
Minimal no
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1728, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9,8]))
 
pari: [g,chi] = znchar(Mod(721,1728))
 

Basic properties

Modulus: \(1728\)
Conductor: \(144\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{144}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1728.bc

\(\chi_{1728}(145,\cdot)\) \(\chi_{1728}(721,\cdot)\) \(\chi_{1728}(1009,\cdot)\) \(\chi_{1728}(1585,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.12.369768517790072832.1

Values on generators

\((703,325,1217)\) → \((1,-i,e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(1\)\(i\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1728 }(721,a) \;\) at \(\;a = \) e.g. 2