# Properties

 Label 1.7_13.12t1.1c4 Dimension 1 Group $C_{12}$ Conductor $7 \cdot 13$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{12}$ Conductor: $91= 7 \cdot 13$ Artin number field: Splitting field of $f= x^{12} - x^{11} - 4 x^{10} - 10 x^{9} + 39 x^{8} - 78 x^{7} + 214 x^{6} - 280 x^{5} + 693 x^{4} - 573 x^{3} - 222 x^{2} + 123 x + 601$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_{12}$ Parity: Odd Corresponding Dirichlet character: $$\chi_{91}(18,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $x^{4} + 23 x + 6$
Roots:
 $r_{ 1 }$ $=$ $38 a^{3} + 24 a^{2} + \left(19 a^{3} + 8 a^{2} + 22 a + 32\right)\cdot 41 + \left(32 a^{3} + 19 a^{2} + 2 a + 38\right)\cdot 41^{2} + \left(20 a^{3} + 16 a^{2} + 5 a + 27\right)\cdot 41^{3} + \left(26 a^{3} + 37 a^{2} + 26 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 2 }$ $=$ $38 a^{3} + 24 a^{2} + 34 + \left(19 a^{3} + 8 a^{2} + 22 a + 29\right)\cdot 41 + \left(32 a^{3} + 19 a^{2} + 2 a + 12\right)\cdot 41^{2} + \left(20 a^{3} + 16 a^{2} + 5 a + 19\right)\cdot 41^{3} + \left(26 a^{3} + 37 a^{2} + 26 a + 16\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 3 }$ $=$ $38 a^{3} + 24 a^{2} + 16 + \left(19 a^{3} + 8 a^{2} + 22 a + 26\right)\cdot 41 + \left(32 a^{3} + 19 a^{2} + 2 a + 30\right)\cdot 41^{2} + \left(20 a^{3} + 16 a^{2} + 5 a + 3\right)\cdot 41^{3} + \left(26 a^{3} + 37 a^{2} + 26 a + 38\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 4 }$ $=$ $16 a^{3} + 34 a^{2} + 20 a + 3 + \left(10 a^{3} + 15 a^{2} + 37 a + 19\right)\cdot 41 + \left(a^{2} + 37 a + 20\right)\cdot 41^{2} + \left(32 a^{3} + 8 a^{2} + 9 a + 38\right)\cdot 41^{3} + \left(7 a^{3} + 20 a^{2} + 36 a\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 5 }$ $=$ $16 a^{3} + 34 a^{2} + 20 a + 10 + \left(10 a^{3} + 15 a^{2} + 37 a + 21\right)\cdot 41 + \left(a^{2} + 37 a + 5\right)\cdot 41^{2} + \left(32 a^{3} + 8 a^{2} + 9 a + 6\right)\cdot 41^{3} + \left(7 a^{3} + 20 a^{2} + 36 a + 10\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 6 }$ $=$ $16 a^{3} + 34 a^{2} + 20 a + 26 + \left(10 a^{3} + 15 a^{2} + 37 a + 15\right)\cdot 41 + \left(a^{2} + 37 a + 38\right)\cdot 41^{2} + \left(32 a^{3} + 8 a^{2} + 9 a + 22\right)\cdot 41^{3} + \left(7 a^{3} + 20 a^{2} + 36 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 7 }$ $=$ $24 a^{3} + 36 a^{2} + 30 a + 25 + \left(13 a^{3} + 12 a^{2} + a + 4\right)\cdot 41 + \left(34 a^{3} + 2 a^{2} + 13 a + 9\right)\cdot 41^{2} + \left(40 a^{2} + 31 a + 32\right)\cdot 41^{3} + \left(8 a^{3} + 36 a^{2} + 8 a + 34\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 8 }$ $=$ $24 a^{3} + 36 a^{2} + 30 a + \left(13 a^{3} + 12 a^{2} + a + 40\right)\cdot 41 + \left(34 a^{3} + 2 a^{2} + 13 a\right)\cdot 41^{2} + \left(40 a^{2} + 31 a + 8\right)\cdot 41^{3} + \left(8 a^{3} + 36 a^{2} + 8 a + 6\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 9 }$ $=$ $24 a^{3} + 36 a^{2} + 30 a + 18 + \left(13 a^{3} + 12 a^{2} + a + 2\right)\cdot 41 + \left(34 a^{3} + 2 a^{2} + 13 a + 24\right)\cdot 41^{2} + \left(40 a^{2} + 31 a + 23\right)\cdot 41^{3} + \left(8 a^{3} + 36 a^{2} + 8 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 10 }$ $=$ $4 a^{3} + 29 a^{2} + 32 a + 1 + \left(38 a^{3} + 3 a^{2} + 20 a + 5\right)\cdot 41 + \left(14 a^{3} + 18 a^{2} + 28 a + 7\right)\cdot 41^{2} + \left(28 a^{3} + 17 a^{2} + 35 a + 37\right)\cdot 41^{3} + \left(39 a^{3} + 28 a^{2} + 10 a + 38\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 11 }$ $=$ $4 a^{3} + 29 a^{2} + 32 a + 8 + \left(38 a^{3} + 3 a^{2} + 20 a + 7\right)\cdot 41 + \left(14 a^{3} + 18 a^{2} + 28 a + 33\right)\cdot 41^{2} + \left(28 a^{3} + 17 a^{2} + 35 a + 4\right)\cdot 41^{3} + \left(39 a^{3} + 28 a^{2} + 10 a + 7\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 12 }$ $=$ $4 a^{3} + 29 a^{2} + 32 a + 24 + \left(38 a^{3} + 3 a^{2} + 20 a + 1\right)\cdot 41 + \left(14 a^{3} + 18 a^{2} + 28 a + 25\right)\cdot 41^{2} + \left(28 a^{3} + 17 a^{2} + 35 a + 21\right)\cdot 41^{3} + \left(39 a^{3} + 28 a^{2} + 10 a + 19\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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