# Properties

 Label 2.100.10t6.b Dimension 2 Group $D_5\times C_5$ Conductor $2^{2} \cdot 5^{2}$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_5\times C_5$ Conductor: $100= 2^{2} \cdot 5^{2}$ Artin number field: Splitting field of $f= x^{10} - 4 x^{9} + 9 x^{8} - 14 x^{7} + 15 x^{6} - 10 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $D_5\times C_5$ Parity: Odd Projective image: $D_5$ Projective field: Galois closure of 5.1.6250000.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 10 }$ Character values $c1$ $c2$ $c3$ $c4$ $1$ $1$ $()$ $2$ $2$ $2$ $2$ $5$ $2$ $(1,2)(3,6)(4,10)(5,9)(7,8)$ $0$ $0$ $0$ $0$ $1$ $5$ $(1,7,6,4,5)(2,8,3,10,9)$ $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ $2 \zeta_{5}^{2}$ $2 \zeta_{5}^{3}$ $2 \zeta_{5}$ $1$ $5$ $(1,6,5,7,4)(2,3,9,8,10)$ $2 \zeta_{5}^{3}$ $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ $2 \zeta_{5}$ $2 \zeta_{5}^{2}$ $1$ $5$ $(1,4,7,5,6)(2,10,8,9,3)$ $2 \zeta_{5}^{2}$ $2 \zeta_{5}$ $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ $2 \zeta_{5}^{3}$ $1$ $5$ $(1,5,4,6,7)(2,9,10,3,8)$ $2 \zeta_{5}$ $2 \zeta_{5}^{3}$ $2 \zeta_{5}^{2}$ $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ $2$ $5$ $(1,7,6,4,5)(2,9,10,3,8)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $2$ $5$ $(1,6,5,7,4)(2,10,8,9,3)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $2$ $5$ $(2,8,3,10,9)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$ $\zeta_{5}^{2} + 1$ $\zeta_{5}^{3} + 1$ $\zeta_{5} + 1$ $2$ $5$ $(2,3,9,8,10)$ $\zeta_{5}^{3} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$ $\zeta_{5} + 1$ $\zeta_{5}^{2} + 1$ $2$ $5$ $(2,10,8,9,3)$ $\zeta_{5}^{2} + 1$ $\zeta_{5} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$ $\zeta_{5}^{3} + 1$ $2$ $5$ $(2,9,10,3,8)$ $\zeta_{5} + 1$ $\zeta_{5}^{3} + 1$ $\zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$ $2$ $5$ $(1,5,4,6,7)(2,3,9,8,10)$ $\zeta_{5}^{3} + \zeta_{5}$ $-\zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{2} + \zeta_{5}$ $-\zeta_{5}^{3} - \zeta_{5} - 1$ $2$ $5$ $(1,4,7,5,6)(2,9,10,3,8)$ $\zeta_{5}^{2} + \zeta_{5}$ $\zeta_{5}^{3} + \zeta_{5}$ $-\zeta_{5}^{3} - \zeta_{5} - 1$ $-\zeta_{5}^{2} - \zeta_{5} - 1$ $2$ $5$ $(1,6,5,7,4)(2,8,3,10,9)$ $-\zeta_{5}^{2} - \zeta_{5} - 1$ $-\zeta_{5}^{3} - \zeta_{5} - 1$ $\zeta_{5}^{3} + \zeta_{5}$ $\zeta_{5}^{2} + \zeta_{5}$ $2$ $5$ $(1,7,6,4,5)(2,10,8,9,3)$ $-\zeta_{5}^{3} - \zeta_{5} - 1$ $\zeta_{5}^{2} + \zeta_{5}$ $-\zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{3} + \zeta_{5}$ $5$ $10$ $(1,8,7,3,6,10,4,9,5,2)$ $0$ $0$ $0$ $0$ $5$ $10$ $(1,3,4,2,7,10,5,8,6,9)$ $0$ $0$ $0$ $0$ $5$ $10$ $(1,9,6,8,5,10,7,2,4,3)$ $0$ $0$ $0$ $0$ $5$ $10$ $(1,2,5,9,4,10,6,3,7,8)$ $0$ $0$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.