# Properties

 Label 2.297.12t18.b Dimension $2$ Group $C_6\times S_3$ Conductor $297$ Indicator $0$

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## Basic invariants

 Dimension: $2$ Group: $C_6\times S_3$ Conductor: $$297$$$$\medspace = 3^{3} \cdot 11$$ Artin number field: Galois closure of 12.0.941480149401.1 Galois orbit size: $2$ Smallest permutation container: $C_6\times S_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.891.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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