# Properties

 Label 2.77.10t6.b.c Dimension 2 Group $D_5\times C_5$ Conductor $7 \cdot 11$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_5\times C_5$ Conductor: $77= 7 \cdot 11$ Artin number field: Splitting field of 10.0.246071287.1 defined by $f= x^{10} - 3 x^{9} + 7 x^{8} - 12 x^{7} + 15 x^{6} - 15 x^{5} + 12 x^{4} - 7 x^{3} + 4 x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $D_5\times C_5$ Parity: Odd Determinant: 1.77.10t1.a.c Projective image: $D_5$ Projective field: Galois closure of 5.1.717409.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $x^{5} + 3 x + 18$
Roots:
 $r_{ 1 }$ $=$ $3 a^{4} + 9 a^{2} + 9 a + 14 + \left(15 a^{4} + 15 a^{3} + 3 a^{2} + 20 a + 13\right)\cdot 23 + \left(22 a^{4} + 15 a^{3} + 16 a^{2} + 9 a + 1\right)\cdot 23^{2} + \left(17 a^{4} + a^{3} + 17 a^{2} + 18 a + 15\right)\cdot 23^{3} + \left(6 a^{4} + a^{3} + 7 a^{2} + 8 a + 19\right)\cdot 23^{4} + \left(22 a^{4} + 12 a^{3} + 18 a^{2} + 11 a + 22\right)\cdot 23^{5} + \left(2 a^{4} + 15 a^{3} + 18 a^{2} + 12 a + 5\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 2 }$ $=$ $4 a^{4} + 20 a^{3} + 16 a^{2} + a + 8 + \left(13 a^{4} + 8 a^{3} + 2 a^{2} + 16 a + 8\right)\cdot 23 + \left(21 a^{4} + 11 a^{3} + 14 a + 3\right)\cdot 23^{2} + \left(3 a^{4} + 3 a^{3} + 10 a^{2} + 14 a + 19\right)\cdot 23^{3} + \left(a^{3} + 4 a^{2} + 19 a + 10\right)\cdot 23^{4} + \left(12 a^{4} + 17 a^{3} + 12 a^{2} + 8 a + 13\right)\cdot 23^{5} + \left(4 a^{4} + 19 a^{3} + 5 a^{2} + 16\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 3 }$ $=$ $6 a^{4} + 21 a^{3} + 16 a^{2} + 6 a + 22 + \left(18 a^{4} + 17 a^{3} + 18 a^{2} + 13 a + 15\right)\cdot 23 + \left(3 a^{4} + 10 a^{3} + 8 a^{2} + 17 a + 6\right)\cdot 23^{2} + \left(3 a^{4} + 10 a^{3} + 10 a^{2} + 12 a + 17\right)\cdot 23^{3} + \left(13 a^{4} + 4 a^{3} + 3 a + 9\right)\cdot 23^{4} + \left(12 a^{4} + 9 a^{3} + 9 a^{2} + 7 a + 19\right)\cdot 23^{5} + \left(12 a^{4} + 16 a^{3} + 20 a^{2} + 8 a + 12\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 4 }$ $=$ $11 a^{4} + 22 a^{3} + 16 a^{2} + 14 a + 11 + \left(3 a^{4} + 17 a^{3} + 4 a^{2} + 13 a + 3\right)\cdot 23 + \left(7 a^{4} + 21 a^{3} + 16 a^{2} + 4 a + 10\right)\cdot 23^{2} + \left(5 a^{4} + 8 a^{3} + 9 a^{2} + 14 a + 22\right)\cdot 23^{3} + \left(10 a^{4} + a^{3} + 2 a^{2} + 11\right)\cdot 23^{4} + \left(22 a^{4} + 10 a^{3} + 16 a^{2} + 13 a + 15\right)\cdot 23^{5} + \left(10 a^{4} + 21 a^{2} + 13\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 5 }$ $=$ $12 a^{4} + 7 a^{3} + 13 a + 8 + \left(5 a^{4} + 21 a^{3} + 13 a^{2} + 9 a + 4\right)\cdot 23 + \left(10 a^{4} + 16 a^{3} + 12 a^{2} + 11 a + 13\right)\cdot 23^{2} + \left(7 a^{4} + 19 a^{3} + 12 a^{2} + 17 a + 3\right)\cdot 23^{3} + \left(13 a^{4} + 8 a^{3} + 11 a^{2} + 8 a + 3\right)\cdot 23^{4} + \left(11 a^{4} + 7 a^{3} + 13 a^{2} + 20\right)\cdot 23^{5} + \left(7 a^{4} + 18 a^{3} + 7 a^{2} + 7 a + 16\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 6 }$ $=$ $12 a^{4} + 15 a^{3} + 3 a^{2} + 9 a + 18 + \left(6 a^{4} + 9 a^{3} + 21 a^{2} + 9 a + 10\right)\cdot 23 + \left(3 a^{4} + 6 a^{3} + 20 a + 5\right)\cdot 23^{2} + \left(a^{4} + 7 a^{3} + 5 a^{2} + 7 a + 3\right)\cdot 23^{3} + \left(4 a^{4} + 17 a^{3} + 11 a^{2} + 20\right)\cdot 23^{4} + \left(14 a^{4} + 2 a^{3} + 8 a^{2} + 15 a + 4\right)\cdot 23^{5} + \left(17 a^{4} + 2 a^{3} + 13 a^{2} + 20 a + 11\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 7 }$ $=$ $13 a^{4} + 14 a^{3} + 18 a^{2} + 16 a + 2 + \left(4 a^{4} + 14 a^{3} + 21 a^{2} + 16 a + 6\right)\cdot 23 + \left(10 a^{4} + 18 a^{3} + 19 a^{2} + 11 a + 8\right)\cdot 23^{2} + \left(9 a^{4} + 15 a^{3} + 10 a^{2} + 19 a + 9\right)\cdot 23^{3} + \left(18 a^{4} + 21 a^{3} + 4 a^{2} + 21 a + 22\right)\cdot 23^{4} + \left(7 a^{4} + 6 a^{3} + a + 7\right)\cdot 23^{5} + \left(7 a^{3} + 8 a^{2} + 16 a + 20\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 8 }$ $=$ $15 a^{4} + 12 a^{3} + 16 a^{2} + 16 a + 6 + \left(8 a^{4} + 8 a^{3} + 6 a^{2} + 5 a + 7\right)\cdot 23 + \left(4 a^{4} + 20 a^{3} + 2 a^{2} + 7 a + 8\right)\cdot 23^{2} + \left(16 a^{4} + 4 a^{3} + a^{2} + 6 a + 15\right)\cdot 23^{3} + \left(7 a^{4} + 21 a^{3} + 21 a^{2} + 22 a + 12\right)\cdot 23^{4} + \left(13 a^{4} + 11 a^{3} + 20 a + 10\right)\cdot 23^{5} + \left(18 a^{4} + 7 a^{3} + 4 a^{2} + a + 20\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 9 }$ $=$ $17 a^{4} + 3 a^{3} + 17 a^{2} + 19 a + 20 + \left(14 a^{4} + 19 a^{3} + 17 a^{2} + 9 a + 21\right)\cdot 23 + \left(19 a^{4} + 6 a^{3} + 16 a^{2} + 19 a + 21\right)\cdot 23^{2} + \left(16 a^{4} + 13 a^{3} + 7 a^{2} + 11 a + 16\right)\cdot 23^{3} + \left(17 a^{4} + 20 a^{2} + 4 a + 13\right)\cdot 23^{4} + \left(15 a^{4} + 19 a^{3} + 21 a^{2} + 11 a + 2\right)\cdot 23^{5} + \left(11 a^{4} + 15 a^{3} + 8 a^{2} + 16 a + 13\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 10 }$ $=$ $22 a^{4} + a^{3} + 4 a^{2} + 12 a + 9 + \left(a^{4} + 5 a^{3} + 5 a^{2}\right)\cdot 23 + \left(12 a^{4} + 9 a^{3} + 21 a^{2} + 21 a + 13\right)\cdot 23^{2} + \left(10 a^{4} + 6 a^{3} + 6 a^{2} + 14 a + 15\right)\cdot 23^{3} + \left(14 a^{3} + 8 a^{2} + a + 13\right)\cdot 23^{4} + \left(6 a^{4} + 18 a^{3} + 14 a^{2} + 2 a + 20\right)\cdot 23^{5} + \left(5 a^{4} + 11 a^{3} + 6 a^{2} + 8 a + 6\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

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

### Character values on conjugacy classes

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