Basic invariants
| Dimension: | $2$ |
| Group: | $C_6\times S_3$ |
| Conductor: | $2592= 2^{5} \cdot 3^{4} $ |
| Artin number field: | Splitting field of $f= x^{12} - 12 x^{10} - 4 x^{9} + 54 x^{8} + 36 x^{7} - 104 x^{6} - 108 x^{5} + 57 x^{4} + 96 x^{3} + 36 x^{2} + 36 x + 33 $ over $\Q$ |
| Size of Galois orbit: | 2 |
| Smallest containing permutation representation: | $C_6\times S_3$ |
| Parity: | Odd |
| Determinant: | 1.2e3_3e2.6t1.4c2 |
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^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2 $
Roots:
| $r_{ 1 }$ | $=$ | $ 3 a^{5} + 4 a^{4} + 8 a^{3} + 9 a^{2} + 4 a + 8 + \left(9 a^{5} + 11 a^{4} + 2 a^{3} + 6 a^{2} + 11\right)\cdot 13 + \left(10 a^{5} + a^{4} + 11 a^{3} + 10 a^{2} + 2 a + 11\right)\cdot 13^{2} + \left(6 a^{5} + 7 a^{4} + 3 a^{3} + 8 a + 12\right)\cdot 13^{3} + \left(9 a^{5} + 12 a^{4} + 10 a^{3} + a^{2} + 4 a + 11\right)\cdot 13^{4} + \left(2 a^{5} + 4 a^{4} + 4 a^{3} + a^{2} + 2 a\right)\cdot 13^{5} + \left(7 a^{5} + 4 a^{4} + 5 a + 9\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 10 a^{5} + 8 a^{4} + 7 a^{3} + 5 a^{2} + a + 12 + \left(6 a^{5} + 8 a^{4} + 9 a^{3} + 2 a^{2} + 9 a + 3\right)\cdot 13 + \left(3 a^{5} + 7 a^{4} + 11 a^{3} + 10 a^{2} + 12 a + 6\right)\cdot 13^{2} + \left(4 a^{5} + 10 a^{4} + 8 a^{3} + 11 a^{2} + 2 a + 7\right)\cdot 13^{3} + \left(7 a^{4} + 3 a^{3} + 11 a + 11\right)\cdot 13^{4} + \left(3 a^{5} + 3 a^{4} + 5 a^{3} + 10 a^{2} + 6\right)\cdot 13^{5} + \left(a^{5} + 7 a^{4} + a^{3} + 5 a^{2} + 11 a + 4\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 4 a^{5} + 8 a^{4} + a^{3} + 5 a^{2} + 5 a + 5 + \left(12 a^{5} + 12 a^{4} + 12 a^{3} + 3 a^{2} + 11 a + 1\right)\cdot 13 + \left(3 a^{5} + 3 a^{4} + 7 a^{3} + 7 a^{2} + a + 3\right)\cdot 13^{2} + \left(7 a^{5} + 5 a^{4} + 10 a^{3} + 6 a^{2} + 11 a + 7\right)\cdot 13^{3} + \left(8 a^{5} + 8 a^{4} + 6 a^{3} + 5 a^{2} + a + 9\right)\cdot 13^{4} + \left(10 a^{5} + 3 a^{4} + 11 a^{3} + 8 a^{2} + 3 a + 8\right)\cdot 13^{5} + \left(7 a^{5} + 6 a^{4} + 8 a^{3} + 5 a^{2} + a + 12\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 4 a^{5} + 6 a^{4} + 11 a^{3} + 6 a + 10 + \left(10 a^{5} + 7 a^{4} + 10 a^{3} + a^{2} + 12 a + 12\right)\cdot 13 + \left(3 a^{5} + 9 a^{4} + 7 a^{3} + 3 a^{2} + 10 a + 11\right)\cdot 13^{2} + \left(9 a^{4} + 6 a^{3} + 12 a^{2} + 5 a\right)\cdot 13^{3} + \left(8 a^{5} + 11 a^{4} + 3 a^{3} + 9 a^{2} + 5 a + 6\right)\cdot 13^{4} + \left(12 a^{5} + 8 a^{4} + 11 a^{3} + 12 a^{2} + 11 a + 5\right)\cdot 13^{5} + \left(7 a^{5} + 5 a^{4} + 5 a^{3} + 9 a^{2} + a + 5\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 5 }$ | $=$ | $ a^{4} + 11 a^{3} + 12 a^{2} + 8 a + 6 + \left(10 a^{5} + 6 a^{4} + 3 a^{2} + 3 a + 10\right)\cdot 13 + \left(11 a^{5} + 3 a^{4} + 3 a^{3} + 5 a^{2} + 11 a + 7\right)\cdot 13^{2} + \left(a^{5} + 8 a^{4} + a + 5\right)\cdot 13^{3} + \left(3 a^{5} + 5 a^{4} + 12 a^{3} + 11 a^{2} + 10 a + 2\right)\cdot 13^{4} + \left(7 a^{5} + 4 a^{4} + 2 a^{3} + a^{2} + 9 a + 5\right)\cdot 13^{5} + \left(4 a^{5} + a^{4} + 11 a^{3} + 7 a^{2} + 9 a + 12\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 4 a^{5} + 4 a^{4} + 9 a^{3} + 9 a^{2} + 12 a + 7 + \left(11 a^{5} + 6 a^{4} + 11 a^{3} + 8 a^{2} + 10 a + 2\right)\cdot 13 + \left(6 a^{5} + 12 a^{2} + 9 a + 8\right)\cdot 13^{2} + \left(a^{5} + 4 a^{4} + 9 a^{3} + 11 a^{2} + 9 a + 6\right)\cdot 13^{3} + \left(12 a^{5} + 7 a^{4} + 7 a^{3} + 10 a^{2} + 12\right)\cdot 13^{4} + \left(3 a^{5} + 9 a^{4} + a^{3} + 4 a^{2} + 8 a + 7\right)\cdot 13^{5} + \left(9 a^{5} + 5 a^{4} + 12 a^{3} + 4 a^{2} + 8 a + 1\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 10 a^{5} + 2 a^{4} + 6 a^{3} + 10 a^{2} + 9 a + 2 + \left(5 a^{4} + 9 a^{3} + 7 a^{2} + 6 a + 1\right)\cdot 13 + \left(3 a^{5} + 3 a^{4} + 12 a^{3} + 2 a^{2} + a + 6\right)\cdot 13^{2} + \left(3 a^{4} + a^{3} + 8 a^{2} + 3 a + 3\right)\cdot 13^{3} + \left(8 a^{5} + a^{4} + 4 a^{3} + 11 a^{2} + 12 a + 10\right)\cdot 13^{4} + \left(8 a^{5} + 10 a^{4} + a^{3} + 4 a^{2} + a + 5\right)\cdot 13^{5} + \left(2 a^{5} + 5 a^{4} + 9 a^{3} + 7 a^{2} + 11 a + 10\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 11 a^{3} + 10 a^{2} + 7 a + 3 + \left(9 a^{5} + 5 a^{4} + 10 a^{3} + 11 a^{2} + 2\right)\cdot 13 + \left(7 a^{5} + 7 a^{4} + a^{3} + 9 a^{2} + 12 a + 8\right)\cdot 13^{2} + \left(8 a^{5} + 3 a^{4} + 10 a^{3} + 9 a^{2} + 2 a + 12\right)\cdot 13^{3} + \left(3 a^{4} + 11 a^{3} + 6 a^{2} + 6 a + 10\right)\cdot 13^{4} + \left(a^{5} + 12 a^{4} + 3 a^{3} + 11 a^{2} + 11 a + 12\right)\cdot 13^{5} + \left(9 a^{5} + 6 a^{4} + 11 a^{3} + 7 a^{2} + 12\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 4 a^{5} + 6 a^{4} + 6 a^{3} + 12 a^{2} + 4 a + 11 + \left(9 a^{3} + 3 a^{2} + 6 a + 5\right)\cdot 13 + \left(5 a^{5} + 5 a^{4} + 3 a^{3} + 4 a^{2} + 7 a + 8\right)\cdot 13^{2} + \left(6 a^{5} + 10 a^{4} + 6 a^{3} + 3 a^{2} + 7 a + 3\right)\cdot 13^{3} + \left(2 a^{4} + 9 a^{3} + 8 a^{2} + 4 a + 5\right)\cdot 13^{4} + \left(3 a^{5} + a^{4} + 11 a^{3} + 3 a^{2} + 2 a + 4\right)\cdot 13^{5} + \left(a^{5} + 6 a^{4} + 8 a^{3} + 5 a^{2} + 5 a + 7\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 5 a^{5} + a^{4} + 3 a^{3} + 12 a^{2} + 9 a + 1 + \left(2 a^{5} + 7 a^{4} + 2 a^{3} + 3 a + 9\right)\cdot 13 + \left(2 a^{5} + 8 a^{4} + 4 a^{3} + 6 a^{2} + a + 1\right)\cdot 13^{2} + \left(4 a^{5} + 3 a^{4} + 6 a^{3} + 7 a^{2} + 5 a + 12\right)\cdot 13^{3} + \left(5 a^{5} + 10 a^{4} + 11 a^{3} + 9 a^{2} + 10 a + 3\right)\cdot 13^{4} + \left(11 a^{5} + 12 a^{4} + 12 a^{3} + 12 a^{2} + a + 9\right)\cdot 13^{5} + \left(8 a^{5} + 4 a^{3} + 2 a^{2} + 3 a + 11\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 3 a^{5} + 11 a^{4} + 9 a^{3} + 6 a^{2} + 10 a + 8 + \left(3 a^{5} + 2 a^{4} + 5 a^{3} + 6 a^{2} + 5 a + 9\right)\cdot 13 + \left(2 a^{5} + 2 a^{4} + 11 a^{3} + 12 a + 11\right)\cdot 13^{2} + \left(4 a^{5} + 6 a^{4} + 8 a^{2} + 6 a + 9\right)\cdot 13^{3} + \left(4 a^{5} + 8 a^{4} + 10 a^{3} + 7 a^{2} + 7 a + 4\right)\cdot 13^{4} + \left(3 a^{5} + 3 a^{4} + 7 a^{3} + 9 a^{2} + 12 a + 7\right)\cdot 13^{5} + \left(a^{5} + 5 a^{3} + 10 a^{2} + 2\right)\cdot 13^{6} +O\left(13^{ 7 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 5 a^{5} + a^{4} + 9 a^{3} + a^{2} + 3 a + 5 + \left(2 a^{5} + 5 a^{4} + 5 a^{3} + 8 a^{2} + 7 a + 7\right)\cdot 13 + \left(4 a^{5} + 11 a^{4} + a^{3} + 5 a^{2} + 7 a + 5\right)\cdot 13^{2} + \left(6 a^{5} + 5 a^{4} + 10 a^{2} + 12 a + 8\right)\cdot 13^{3} + \left(4 a^{5} + 11 a^{4} + 7 a^{2} + 2 a + 1\right)\cdot 13^{4} + \left(10 a^{5} + 2 a^{4} + 3 a^{3} + 9 a^{2} + 12 a + 3\right)\cdot 13^{5} + \left(3 a^{5} + a^{4} + 11 a^{3} + 10 a^{2} + 5 a\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 value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,11)(5,10)(7,9)(8,12)$ | $-2$ |
| $3$ | $2$ | $(1,12)(2,4)(3,8)(5,9)(6,11)(7,10)$ | $0$ |
| $3$ | $2$ | $(1,8)(2,11)(3,12)(4,6)(5,7)(9,10)$ | $0$ |
| $1$ | $3$ | $(1,2,5)(3,6,10)(4,9,12)(7,8,11)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,5,2)(3,10,6)(4,12,9)(7,11,8)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,2,5)(3,6,10)(4,12,9)(7,11,8)$ | $-1$ |
| $2$ | $3$ | $(4,9,12)(7,8,11)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(4,12,9)(7,11,8)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,10,2,3,5,6)(4,8,9,11,12,7)$ | $2 \zeta_{3} + 2$ |
| $1$ | $6$ | $(1,6,5,3,2,10)(4,7,12,11,9,8)$ | $-2 \zeta_{3}$ |
| $2$ | $6$ | $(1,6,5,3,2,10)(4,8,9,11,12,7)$ | $1$ |
| $2$ | $6$ | $(1,3)(2,6)(4,7,12,11,9,8)(5,10)$ | $-\zeta_{3} - 1$ |
| $2$ | $6$ | $(1,3)(2,6)(4,8,9,11,12,7)(5,10)$ | $\zeta_{3}$ |
| $3$ | $6$ | $(1,9,2,12,5,4)(3,7,6,8,10,11)$ | $0$ |
| $3$ | $6$ | $(1,4,5,12,2,9)(3,11,10,8,6,7)$ | $0$ |
| $3$ | $6$ | $(1,7,2,8,5,11)(3,9,6,12,10,4)$ | $0$ |
| $3$ | $6$ | $(1,11,5,8,2,7)(3,4,10,12,6,9)$ | $0$ |