Properties

 Label 2.2300.12t11.a Dimension $2$ Group $S_3 \times C_4$ Conductor $2300$ Indicator $0$

Related objects

Basic invariants

 Dimension: $2$ Group: $S_3 \times C_4$ Conductor: $$2300$$$$\medspace = 2^{2} \cdot 5^{2} \cdot 23$$ Artin number field: Galois closure of 12.0.139920500000000.1 Galois orbit size: $2$ Smallest permutation container: $S_3 \times C_4$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.460.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{4} + 3x^{2} + 12x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$5 a^{3} + 8 a^{2} + 4 a + 11 + \left(12 a^{3} + 6 a^{2} + 9 a + 2\right)\cdot 13 + \left(5 a^{3} + 3 a^{2} + 4 a + 2\right)\cdot 13^{2} + \left(4 a^{3} + 10 a^{2} + 9 a + 2\right)\cdot 13^{3} + \left(a^{3} + 6 a^{2}\right)\cdot 13^{4} + \left(12 a^{3} + 5 a^{2} + 11 a + 1\right)\cdot 13^{5} + \left(10 a^{3} + 3 a^{2} + a + 5\right)\cdot 13^{6} + \left(4 a^{3} + 12 a^{2} + 5 a + 5\right)\cdot 13^{7} +O(13^{8})$$ 5*a^3 + 8*a^2 + 4*a + 11 + (12*a^3 + 6*a^2 + 9*a + 2)*13 + (5*a^3 + 3*a^2 + 4*a + 2)*13^2 + (4*a^3 + 10*a^2 + 9*a + 2)*13^3 + (a^3 + 6*a^2)*13^4 + (12*a^3 + 5*a^2 + 11*a + 1)*13^5 + (10*a^3 + 3*a^2 + a + 5)*13^6 + (4*a^3 + 12*a^2 + 5*a + 5)*13^7+O(13^8) $r_{ 2 }$ $=$ $$11 a^{3} + 3 a^{2} + 2 a + 1 + \left(2 a^{3} + 5 a^{2} + 10 a + 11\right)\cdot 13 + \left(12 a^{3} + 2 a^{2} + 5 a + 11\right)\cdot 13^{2} + \left(11 a^{3} + 9 a^{2} + 11 a + 9\right)\cdot 13^{3} + \left(12 a^{3} + 9 a^{2} + a\right)\cdot 13^{4} + \left(9 a^{3} + a^{2} + 12 a + 8\right)\cdot 13^{5} + \left(10 a^{3} + 6 a^{2} + 11 a + 9\right)\cdot 13^{6} + \left(9 a^{3} + 8 a^{2} + 3\right)\cdot 13^{7} +O(13^{8})$$ 11*a^3 + 3*a^2 + 2*a + 1 + (2*a^3 + 5*a^2 + 10*a + 11)*13 + (12*a^3 + 2*a^2 + 5*a + 11)*13^2 + (11*a^3 + 9*a^2 + 11*a + 9)*13^3 + (12*a^3 + 9*a^2 + a)*13^4 + (9*a^3 + a^2 + 12*a + 8)*13^5 + (10*a^3 + 6*a^2 + 11*a + 9)*13^6 + (9*a^3 + 8*a^2 + 3)*13^7+O(13^8) $r_{ 3 }$ $=$ $$9 a^{3} + a + 12 + \left(6 a^{3} + 9 a^{2} + 6 a + 9\right)\cdot 13 + \left(5 a^{2} + 3 a + 5\right)\cdot 13^{2} + \left(3 a^{3} + 4 a^{2} + 3 a\right)\cdot 13^{3} + \left(12 a^{3} + 9 a^{2} + 10 a\right)\cdot 13^{4} + \left(8 a^{3} + 3 a^{2} + 4\right)\cdot 13^{5} + \left(12 a^{3} + 11 a^{2} + 11 a + 4\right)\cdot 13^{6} + \left(4 a^{3} + 3 a^{2} + 6 a + 6\right)\cdot 13^{7} +O(13^{8})$$ 9*a^3 + a + 12 + (6*a^3 + 9*a^2 + 6*a + 9)*13 + (5*a^2 + 3*a + 5)*13^2 + (3*a^3 + 4*a^2 + 3*a)*13^3 + (12*a^3 + 9*a^2 + 10*a)*13^4 + (8*a^3 + 3*a^2 + 4)*13^5 + (12*a^3 + 11*a^2 + 11*a + 4)*13^6 + (4*a^3 + 3*a^2 + 6*a + 6)*13^7+O(13^8) $r_{ 4 }$ $=$ $$5 a^{3} + 10 a + 1 + \left(4 a^{3} + 7 a^{2} + a + 8\right)\cdot 13 + \left(a^{3} + 2 a^{2} + 11 a + 11\right)\cdot 13^{2} + \left(a^{3} + 10 a^{2} + 4 a + 4\right)\cdot 13^{3} + \left(6 a^{3} + a^{2} + 3 a + 5\right)\cdot 13^{4} + \left(9 a^{2} + 6 a + 11\right)\cdot 13^{5} + \left(3 a^{3} + 10 a^{2} + 8 a + 11\right)\cdot 13^{6} + \left(2 a^{3} + 2 a^{2} + 2 a + 4\right)\cdot 13^{7} +O(13^{8})$$ 5*a^3 + 10*a + 1 + (4*a^3 + 7*a^2 + a + 8)*13 + (a^3 + 2*a^2 + 11*a + 11)*13^2 + (a^3 + 10*a^2 + 4*a + 4)*13^3 + (6*a^3 + a^2 + 3*a + 5)*13^4 + (9*a^2 + 6*a + 11)*13^5 + (3*a^3 + 10*a^2 + 8*a + 11)*13^6 + (2*a^3 + 2*a^2 + 2*a + 4)*13^7+O(13^8) $r_{ 5 }$ $=$ $$3 a^{3} + 10 a^{2} + 7 a + 9 + \left(8 a^{3} + 8 a^{2} + 6 a + 7\right)\cdot 13 + \left(2 a^{3} + 11 a^{2} + a + 10\right)\cdot 13^{2} + \left(10 a^{3} + 7 a^{2} + 7 a + 4\right)\cdot 13^{3} + \left(3 a^{3} + a + 6\right)\cdot 13^{4} + \left(a^{3} + 2 a^{2} + 11 a + 2\right)\cdot 13^{5} + \left(8 a^{3} + 9 a^{2} + 4 a + 1\right)\cdot 13^{6} + \left(11 a^{3} + 2 a^{2} + 6 a\right)\cdot 13^{7} +O(13^{8})$$ 3*a^3 + 10*a^2 + 7*a + 9 + (8*a^3 + 8*a^2 + 6*a + 7)*13 + (2*a^3 + 11*a^2 + a + 10)*13^2 + (10*a^3 + 7*a^2 + 7*a + 4)*13^3 + (3*a^3 + a + 6)*13^4 + (a^3 + 2*a^2 + 11*a + 2)*13^5 + (8*a^3 + 9*a^2 + 4*a + 1)*13^6 + (11*a^3 + 2*a^2 + 6*a)*13^7+O(13^8) $r_{ 6 }$ $=$ $$7 a^{3} + a^{2} + 10 a + 2 + \left(8 a^{3} + 3 a^{2} + 12 a + 11\right)\cdot 13 + \left(12 a^{3} + 2 a^{2} + 8 a + 5\right)\cdot 13^{2} + \left(2 a^{3} + 11 a^{2} + 3\right)\cdot 13^{3} + \left(10 a^{2} + 8 a + 11\right)\cdot 13^{4} + \left(10 a^{3} + 10 a^{2} + 2 a + 4\right)\cdot 13^{5} + \left(4 a^{3} + 12 a^{2} + 8 a\right)\cdot 13^{6} + \left(10 a^{3} + a^{2} + 5 a\right)\cdot 13^{7} +O(13^{8})$$ 7*a^3 + a^2 + 10*a + 2 + (8*a^3 + 3*a^2 + 12*a + 11)*13 + (12*a^3 + 2*a^2 + 8*a + 5)*13^2 + (2*a^3 + 11*a^2 + 3)*13^3 + (10*a^2 + 8*a + 11)*13^4 + (10*a^3 + 10*a^2 + 2*a + 4)*13^5 + (4*a^3 + 12*a^2 + 8*a)*13^6 + (10*a^3 + a^2 + 5*a)*13^7+O(13^8) $r_{ 7 }$ $=$ $$12 a^{3} + 2 a^{2} + 6 a + 3 + \left(7 a^{3} + 4 a^{2} + 10 a + 7\right)\cdot 13 + \left(9 a^{3} + 12 a^{2} + 7 a + 6\right)\cdot 13^{2} + \left(7 a^{3} + 8 a^{2} + 10 a + 10\right)\cdot 13^{3} + \left(5 a^{3} + 8 a^{2} + 3 a + 10\right)\cdot 13^{4} + \left(9 a^{2} + 11 a + 7\right)\cdot 13^{5} + \left(6 a^{3} + 11 a^{2} + 10 a + 3\right)\cdot 13^{6} + \left(3 a^{3} + 2 a^{2} + 7 a + 4\right)\cdot 13^{7} +O(13^{8})$$ 12*a^3 + 2*a^2 + 6*a + 3 + (7*a^3 + 4*a^2 + 10*a + 7)*13 + (9*a^3 + 12*a^2 + 7*a + 6)*13^2 + (7*a^3 + 8*a^2 + 10*a + 10)*13^3 + (5*a^3 + 8*a^2 + 3*a + 10)*13^4 + (9*a^2 + 11*a + 7)*13^5 + (6*a^3 + 11*a^2 + 10*a + 3)*13^6 + (3*a^3 + 2*a^2 + 7*a + 4)*13^7+O(13^8) $r_{ 8 }$ $=$ $$3 a^{3} + 7 a^{2} + 3 a + 11 + \left(8 a^{3} + a^{2} + 9 a + 9\right)\cdot 13 + \left(10 a^{3} + 2 a^{2} + 7 a + 9\right)\cdot 13^{2} + \left(6 a^{3} + a + 1\right)\cdot 13^{3} + \left(6 a^{3} + 4 a^{2} + 10 a + 10\right)\cdot 13^{4} + \left(a^{3} + 6 a^{2} + 10\right)\cdot 13^{5} + \left(a^{3} + a^{2} + 7 a + 4\right)\cdot 13^{6} + \left(9 a^{3} + 12 a^{2} + 2 a + 4\right)\cdot 13^{7} +O(13^{8})$$ 3*a^3 + 7*a^2 + 3*a + 11 + (8*a^3 + a^2 + 9*a + 9)*13 + (10*a^3 + 2*a^2 + 7*a + 9)*13^2 + (6*a^3 + a + 1)*13^3 + (6*a^3 + 4*a^2 + 10*a + 10)*13^4 + (a^3 + 6*a^2 + 10)*13^5 + (a^3 + a^2 + 7*a + 4)*13^6 + (9*a^3 + 12*a^2 + 2*a + 4)*13^7+O(13^8) $r_{ 9 }$ $=$ $$2 a^{3} + a^{2} + 12 a + 6 + \left(10 a^{3} + 9 a^{2} + 12\right)\cdot 13 + \left(6 a^{3} + 8 a^{2} + 12 a + 10\right)\cdot 13^{2} + \left(4 a^{3} + 7 a^{2} + 7 a + 11\right)\cdot 13^{3} + \left(a^{3} + a^{2} + 11\right)\cdot 13^{4} + \left(11 a^{3} + 12 a^{2} + 3 a + 1\right)\cdot 13^{5} + \left(5 a^{3} + 11 a^{2} + 12 a + 11\right)\cdot 13^{6} + \left(11 a^{2} + 11 a + 10\right)\cdot 13^{7} +O(13^{8})$$ 2*a^3 + a^2 + 12*a + 6 + (10*a^3 + 9*a^2 + 12)*13 + (6*a^3 + 8*a^2 + 12*a + 10)*13^2 + (4*a^3 + 7*a^2 + 7*a + 11)*13^3 + (a^3 + a^2 + 11)*13^4 + (11*a^3 + 12*a^2 + 3*a + 1)*13^5 + (5*a^3 + 11*a^2 + 12*a + 11)*13^6 + (11*a^2 + 11*a + 10)*13^7+O(13^8) $r_{ 10 }$ $=$ $$4 a^{3} + 12 a^{2} + 6 a + 10 + \left(2 a^{3} + 2 a^{2} + 5 a + 2\right)\cdot 13 + 4\cdot 13^{2} + \left(8 a^{3} + a^{2} + 10 a + 1\right)\cdot 13^{3} + \left(10 a^{3} + 10 a^{2}\right)\cdot 13^{4} + \left(3 a^{3} + a^{2} + 3 a + 11\right)\cdot 13^{5} + \left(12 a^{3} + 6 a^{2} + 11 a + 3\right)\cdot 13^{6} + \left(9 a^{3} + 8 a^{2} + 12 a + 11\right)\cdot 13^{7} +O(13^{8})$$ 4*a^3 + 12*a^2 + 6*a + 10 + (2*a^3 + 2*a^2 + 5*a + 2)*13 + 4*13^2 + (8*a^3 + a^2 + 10*a + 1)*13^3 + (10*a^3 + 10*a^2)*13^4 + (3*a^3 + a^2 + 3*a + 11)*13^5 + (12*a^3 + 6*a^2 + 11*a + 3)*13^6 + (9*a^3 + 8*a^2 + 12*a + 11)*13^7+O(13^8) $r_{ 11 }$ $=$ $$11 a^{3} + 10 a^{2} + 9 a + 6 + \left(2 a^{3} + 9 a^{2} + 9 a + 2\right)\cdot 13 + \left(3 a^{3} + 5 a^{2} + 5 a + 4\right)\cdot 13^{2} + \left(12 a^{3} + a^{2} + 11 a + 7\right)\cdot 13^{3} + \left(7 a^{3} + 10 a^{2} + 3 a + 1\right)\cdot 13^{4} + \left(6 a^{3} + a^{2} + 11 a + 6\right)\cdot 13^{5} + \left(2 a^{3} + 3 a^{2} + 8 a + 4\right)\cdot 13^{6} + \left(7 a^{3} + 4 a^{2} + 5 a + 7\right)\cdot 13^{7} +O(13^{8})$$ 11*a^3 + 10*a^2 + 9*a + 6 + (2*a^3 + 9*a^2 + 9*a + 2)*13 + (3*a^3 + 5*a^2 + 5*a + 4)*13^2 + (12*a^3 + a^2 + 11*a + 7)*13^3 + (7*a^3 + 10*a^2 + 3*a + 1)*13^4 + (6*a^3 + a^2 + 11*a + 6)*13^5 + (2*a^3 + 3*a^2 + 8*a + 4)*13^6 + (7*a^3 + 4*a^2 + 5*a + 7)*13^7+O(13^8) $r_{ 12 }$ $=$ $$6 a^{3} + 11 a^{2} + 8 a + 7 + \left(3 a^{3} + 10 a^{2} + 8 a + 5\right)\cdot 13 + \left(12 a^{3} + 7 a^{2} + 8 a + 7\right)\cdot 13^{2} + \left(4 a^{3} + 5 a^{2} + 12 a + 6\right)\cdot 13^{3} + \left(9 a^{3} + 4 a^{2} + 6 a + 6\right)\cdot 13^{4} + \left(11 a^{3} + 4 a + 8\right)\cdot 13^{5} + \left(12 a^{3} + 3 a^{2} + 7 a + 4\right)\cdot 13^{6} + \left(3 a^{3} + 6 a^{2} + 9 a + 6\right)\cdot 13^{7} +O(13^{8})$$ 6*a^3 + 11*a^2 + 8*a + 7 + (3*a^3 + 10*a^2 + 8*a + 5)*13 + (12*a^3 + 7*a^2 + 8*a + 7)*13^2 + (4*a^3 + 5*a^2 + 12*a + 6)*13^3 + (9*a^3 + 4*a^2 + 6*a + 6)*13^4 + (11*a^3 + 4*a + 8)*13^5 + (12*a^3 + 3*a^2 + 7*a + 4)*13^6 + (3*a^3 + 6*a^2 + 9*a + 6)*13^7+O(13^8)

Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

 Cycle notation $(1,11)(2,5)(3,7)(4,10)(6,8)(9,12)$ $(2,7)(3,12)(4,6)(10,11)$ $(1,7,8,3)(2,11,12,6)(4,9,10,5)$ $(1,10)(2,12)(3,9)(4,8)(5,7)(6,11)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 12 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,8)(2,12)(3,7)(4,10)(5,9)(6,11)$ $-2$ $-2$ $3$ $2$ $(1,11)(2,5)(3,7)(4,10)(6,8)(9,12)$ $0$ $0$ $3$ $2$ $(2,7)(3,12)(4,6)(10,11)$ $0$ $0$ $2$ $3$ $(1,6,4)(2,7,9)(3,5,12)(8,11,10)$ $-1$ $-1$ $1$ $4$ $(1,9,8,5)(2,11,12,6)(3,4,7,10)$ $-2 \zeta_{4}$ $2 \zeta_{4}$ $1$ $4$ $(1,5,8,9)(2,6,12,11)(3,10,7,4)$ $2 \zeta_{4}$ $-2 \zeta_{4}$ $3$ $4$ $(1,7,8,3)(2,11,12,6)(4,9,10,5)$ $0$ $0$ $3$ $4$ $(1,3,8,7)(2,6,12,11)(4,5,10,9)$ $0$ $0$ $2$ $6$ $(1,10,6,8,4,11)(2,5,7,12,9,3)$ $1$ $1$ $2$ $12$ $(1,12,10,9,6,3,8,2,4,5,11,7)$ $-\zeta_{4}$ $\zeta_{4}$ $2$ $12$ $(1,2,10,5,6,7,8,12,4,9,11,3)$ $\zeta_{4}$ $-\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.