Basic invariants
| Dimension: | $2$ |
| Group: | $S_3 \times C_4$ |
| Conductor: | $675= 3^{3} \cdot 5^{2} $ |
| Artin number field: | Splitting field of $f= x^{12} - 6 x^{11} + 15 x^{10} - 11 x^{9} - 27 x^{8} + 69 x^{7} - 40 x^{6} - 45 x^{5} + 57 x^{4} + 8 x^{3} - 24 x^{2} - 3 x + 1 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $S_3 \times C_4$ |
| Parity: | Odd |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{4} + 3 x^{2} + 12 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 6 a^{3} + 9 a^{2} + 2 a + 8 + \left(a^{3} + 5 a^{2} + 6 a + 10\right)\cdot 13 + \left(5 a^{3} + 6 a^{2} + a + 2\right)\cdot 13^{2} + \left(a^{3} + 11 a^{2} + 8 a + 5\right)\cdot 13^{3} + \left(4 a^{3} + a^{2} + 2 a + 5\right)\cdot 13^{4} + \left(a^{2} + 10\right)\cdot 13^{5} + \left(3 a^{3} + 12 a^{2} + 8 a + 2\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 4 a^{3} + 2 a^{2} + 2 a + 12 + \left(7 a^{3} + 10 a^{2} + 12 a + 9\right)\cdot 13 + \left(9 a^{3} + 7 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(a^{3} + 5 a^{2} + 10 a + 2\right)\cdot 13^{3} + \left(a^{3} + 3 a^{2} + 4 a + 4\right)\cdot 13^{4} + \left(9 a^{3} + 11 a^{2} + 12 a + 6\right)\cdot 13^{5} + \left(a^{3} + a^{2} + 3 a + 8\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 10 a^{3} + 2 a^{2} + a + 1 + \left(4 a^{3} + a^{2} + 12 a + 1\right)\cdot 13 + \left(9 a^{3} + 2 a^{2} + 7 a + 2\right)\cdot 13^{2} + \left(6 a^{3} + 11 a^{2} + 5 a + 7\right)\cdot 13^{3} + \left(2 a^{3} + 7 a^{2} + 7 a + 6\right)\cdot 13^{4} + \left(4 a^{3} + 11 a^{2} + 11 a + 2\right)\cdot 13^{5} + \left(12 a^{3} + 4 a^{2} + 9 a + 4\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 2 a^{3} + 5 a^{2} + 3 a + 5 + \left(11 a^{3} + 9 a^{2} + a + 4\right)\cdot 13 + \left(8 a^{3} + 10 a^{2} + a + 9\right)\cdot 13^{2} + \left(12 a^{3} + 7 a^{2} + a + 6\right)\cdot 13^{3} + \left(7 a^{3} + 6 a^{2} + 9 a + 5\right)\cdot 13^{4} + \left(2 a^{3} + 6 a^{2} + 11 a + 12\right)\cdot 13^{5} + \left(3 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 5 }$ | $=$ | $ a^{3} + 11 a^{2} + 3 + \left(4 a^{2} + 7 a + 12\right)\cdot 13 + \left(a^{3} + 3 a^{2} + 11 a + 3\right)\cdot 13^{2} + \left(2 a^{3} + a^{2} + 2 a + 8\right)\cdot 13^{3} + \left(10 a^{3} + 7 a^{2} + 8 a + 4\right)\cdot 13^{4} + \left(6 a^{2} + a + 12\right)\cdot 13^{5} + \left(8 a^{3} + 6 a^{2} + 6 a + 10\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 9 a^{3} + 11 a^{2} + 11 a + 12 + \left(2 a^{3} + 11 a^{2} + 12 a + 9\right)\cdot 13 + \left(12 a^{3} + 7 a^{2} + 2\right)\cdot 13^{2} + \left(10 a^{3} + 3 a^{2} + 6 a + 4\right)\cdot 13^{3} + \left(8 a^{3} + 8 a^{2} + 7 a + 9\right)\cdot 13^{4} + \left(5 a^{3} + 3 a^{2} + a + 9\right)\cdot 13^{5} + \left(4 a^{3} + 9 a^{2} + 11 a + 11\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 2 a^{3} + 6 a^{2} + a + 11 + \left(7 a^{3} + 9 a^{2} + 10 a + 4\right)\cdot 13 + \left(11 a^{3} + 12 a^{2} + 7 a + 2\right)\cdot 13^{2} + \left(12 a^{3} + 12 a^{2} + 9 a + 6\right)\cdot 13^{3} + \left(9 a + 10\right)\cdot 13^{4} + \left(11 a^{3} + 10 a^{2} + 6 a + 12\right)\cdot 13^{5} + \left(12 a^{3} + 7 a^{2} + a + 3\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 6 a^{3} + 2 a^{2} + 4 + \left(5 a^{3} + 4 a^{2} + 6 a + 5\right)\cdot 13 + \left(7 a^{3} + 11 a^{2} + a + 11\right)\cdot 13^{2} + \left(8 a^{3} + 6 a^{2} + 3 a + 10\right)\cdot 13^{3} + \left(11 a^{3} + a^{2} + 11 a + 7\right)\cdot 13^{4} + \left(5 a^{3} + 3 a^{2} + 8 a + 11\right)\cdot 13^{5} + \left(2 a^{3} + 11 a^{2} + 8 a + 2\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 3 a^{3} + 7 a^{2} + 10 a + 2 + \left(8 a^{3} + 10 a^{2} + 7 a + 3\right)\cdot 13 + \left(12 a^{3} + 8 a^{2} + 7 a + 6\right)\cdot 13^{2} + \left(10 a^{3} + a^{2} + 10 a + 4\right)\cdot 13^{3} + \left(a^{3} + 8 a^{2} + 3 a + 9\right)\cdot 13^{4} + \left(3 a^{3} + 2 a^{2} + 10 a + 1\right)\cdot 13^{5} + \left(2 a^{2} + 12\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 7 a^{3} + 2 a^{2} + 2 a + 11 + \left(10 a^{3} + a^{2} + a + 9\right)\cdot 13 + \left(a^{2} + 12 a + 11\right)\cdot 13^{2} + \left(10 a^{2} + 2 a + 9\right)\cdot 13^{3} + \left(9 a^{2} + 4 a + 1\right)\cdot 13^{4} + \left(11 a^{3} + 6 a^{2} + 7 a + 7\right)\cdot 13^{5} + \left(4 a^{3} + 9 a^{2} + 4 a + 12\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 11 a^{3} + 8 a^{2} + 10 a + 6 + \left(4 a^{3} + 7 a^{2} + 12 a + 3\right)\cdot 13 + \left(8 a^{3} + 12 a^{2} + a + 1\right)\cdot 13^{2} + \left(8 a^{2} + 8 a + 4\right)\cdot 13^{3} + \left(8 a^{3} + 7 a^{2} + 4 a + 1\right)\cdot 13^{4} + \left(8 a^{3} + 4 a^{2} + 5\right)\cdot 13^{5} + \left(3 a^{3} + 9 a^{2} + a + 11\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 4 a^{3} + 10 a + 9 + \left(a^{3} + 2 a^{2} + a + 3\right)\cdot 13 + \left(4 a^{3} + 6 a^{2} + 2 a + 6\right)\cdot 13^{2} + \left(9 a^{3} + 9 a^{2} + 9 a + 8\right)\cdot 13^{3} + \left(7 a^{3} + a^{2} + 4 a + 11\right)\cdot 13^{4} + \left(2 a^{3} + 10 a^{2} + 5 a + 11\right)\cdot 13^{5} + \left(8 a^{3} + 10 a^{2} + 12 a + 1\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
Generators of the action on the roots $r_1, \ldots, r_{ 12 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 12 }$ | Character values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,12)(2,6)(3,8)(4,11)(5,7)(9,10)$ | $-2$ | $-2$ |
| $3$ | $2$ | $(1,12)(2,8)(3,6)(4,5)(7,11)(9,10)$ | $0$ | $0$ |
| $3$ | $2$ | $(1,11)(2,10)(4,12)(6,9)$ | $0$ | $0$ |
| $2$ | $3$ | $(1,5,11)(2,10,3)(4,12,7)(6,9,8)$ | $-1$ | $-1$ |
| $1$ | $4$ | $(1,9,12,10)(2,11,6,4)(3,5,8,7)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,10,12,9)(2,4,6,11)(3,7,8,5)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $3$ | $4$ | $(1,6,12,2)(3,5,8,7)(4,10,11,9)$ | $0$ | $0$ |
| $3$ | $4$ | $(1,2,12,6)(3,7,8,5)(4,9,11,10)$ | $0$ | $0$ |
| $2$ | $6$ | $(1,4,5,12,11,7)(2,8,10,6,3,9)$ | $1$ | $1$ |
| $2$ | $12$ | $(1,2,7,9,11,3,12,6,5,10,4,8)$ | $-\zeta_{4}$ | $\zeta_{4}$ |
| $2$ | $12$ | $(1,6,7,10,11,8,12,2,5,9,4,3)$ | $\zeta_{4}$ | $-\zeta_{4}$ |