# Properties

 Label 1.100.10t1.a.b Dimension 1 Group $C_{10}$ Conductor $2^{2} \cdot 5^{2}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{10}$ Conductor: $100= 2^{2} \cdot 5^{2}$ Artin number field: Splitting field of 10.0.156250000000000.1 defined by $f= x^{10} + 20 x^{8} + 120 x^{6} + 225 x^{4} + 90 x^{2} + 1$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_{10}$ Parity: Odd Corresponding Dirichlet character: $$\chi_{100}(31,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## 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:
 $r_{ 1 }$ $=$ $4 a^{4} + a^{3} + 2 a^{2} + 6 a + 5 + \left(10 a^{4} + 4 a^{3} + 8 a^{2} + 3 a + 4\right)\cdot 13 + \left(6 a^{3} + 7 a^{2} + 2 a + 10\right)\cdot 13^{2} + \left(a^{4} + 2 a^{3} + 12 a^{2} + 7 a\right)\cdot 13^{3} + \left(a^{4} + 8 a^{3} + 10 a^{2} + 4 a + 6\right)\cdot 13^{4} + \left(4 a^{4} + 8 a^{3} + 6 a^{2} + 3 a + 10\right)\cdot 13^{5} + \left(5 a^{4} + 5 a^{3} + 10 a^{2} + 5 a + 11\right)\cdot 13^{6} + \left(8 a^{4} + 3 a^{3} + a\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 2 }$ $=$ $9 a^{4} + 10 a^{3} + 8 a^{2} + 8 + \left(2 a^{4} + 11 a^{3} + a^{2} + 5 a + 8\right)\cdot 13 + \left(7 a^{4} + 7 a^{3} + 6 a^{2} + 4 a + 12\right)\cdot 13^{2} + \left(12 a^{4} + 6 a^{3} + 7 a^{2} + 2 a + 8\right)\cdot 13^{3} + \left(7 a^{4} + 4 a^{3} + 2 a + 4\right)\cdot 13^{4} + \left(12 a^{4} + 3 a^{3} + 6 a^{2} + 6 a + 1\right)\cdot 13^{5} + \left(4 a^{4} + 2 a^{3} + 12 a^{2} + 4 a + 8\right)\cdot 13^{6} + \left(3 a^{4} + 4 a^{2} + 11 a + 5\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 3 }$ $=$ $3 a^{4} + 10 a^{3} + 4 a + 7 + \left(9 a^{4} + 10 a^{3} + 7 a^{2} + 7 a + 3\right)\cdot 13 + \left(11 a^{4} + 6 a^{3} + 2 a^{2} + 2 a + 6\right)\cdot 13^{2} + \left(5 a^{4} + 6 a^{3} + 6 a^{2} + 5 a + 8\right)\cdot 13^{3} + \left(6 a^{4} + 8 a^{3} + 5 a^{2} + 7\right)\cdot 13^{4} + \left(6 a^{4} + 3 a^{3} + 11 a^{2} + 4 a + 2\right)\cdot 13^{5} + \left(a^{4} + 7 a^{3} + 3 a^{2} + 9 a + 2\right)\cdot 13^{6} + \left(6 a^{4} + 12 a^{3} + a^{2} + 11 a + 9\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 4 }$ $=$ $11 a^{4} + 3 a^{2} + 4 + \left(10 a^{4} + 12 a^{3} + 2 a^{2} + 6 a + 11\right)\cdot 13 + \left(3 a^{4} + 7 a^{3} + 11 a^{2} + 5 a + 1\right)\cdot 13^{2} + \left(12 a^{3} + a^{2} + 5 a + 6\right)\cdot 13^{3} + \left(4 a^{4} + 3 a^{3} + 3 a^{2} + 2\right)\cdot 13^{4} + \left(11 a^{4} + 5 a^{3} + 7 a^{2} + 4 a + 10\right)\cdot 13^{5} + \left(a^{4} + 10 a^{3} + 9 a^{2} + 11 a\right)\cdot 13^{6} + \left(12 a^{4} + 6 a^{3} + 7 a^{2} + 4 a + 5\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 5 }$ $=$ $5 a^{4} + 8 a^{3} + 7 a^{2} + 10 a + 3 + \left(11 a^{4} + a^{3} + a^{2} + 9 a + 5\right)\cdot 13 + \left(2 a^{4} + 5 a^{2} + 3 a + 1\right)\cdot 13^{2} + \left(6 a^{4} + 3 a^{3} + 11 a^{2} + 9 a + 12\right)\cdot 13^{3} + \left(11 a^{4} + 4 a^{3} + 6 a + 2\right)\cdot 13^{4} + \left(11 a^{4} + 10 a^{3} + 4 a^{2} + 9 a + 4\right)\cdot 13^{5} + \left(9 a^{4} + 4 a^{3} + 4 a^{2} + 7 a + 8\right)\cdot 13^{6} + \left(5 a^{4} + 9 a^{3} + 12 a^{2} + 6 a + 10\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 6 }$ $=$ $9 a^{4} + 12 a^{3} + 11 a^{2} + 7 a + 8 + \left(2 a^{4} + 8 a^{3} + 4 a^{2} + 9 a + 8\right)\cdot 13 + \left(12 a^{4} + 6 a^{3} + 5 a^{2} + 10 a + 2\right)\cdot 13^{2} + \left(11 a^{4} + 10 a^{3} + 5 a + 12\right)\cdot 13^{3} + \left(11 a^{4} + 4 a^{3} + 2 a^{2} + 8 a + 6\right)\cdot 13^{4} + \left(8 a^{4} + 4 a^{3} + 6 a^{2} + 9 a + 2\right)\cdot 13^{5} + \left(7 a^{4} + 7 a^{3} + 2 a^{2} + 7 a + 1\right)\cdot 13^{6} + \left(4 a^{4} + 9 a^{3} + 12 a^{2} + 11 a + 12\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 7 }$ $=$ $4 a^{4} + 3 a^{3} + 5 a^{2} + 5 + \left(10 a^{4} + a^{3} + 11 a^{2} + 8 a + 4\right)\cdot 13 + \left(5 a^{4} + 5 a^{3} + 6 a^{2} + 8 a\right)\cdot 13^{2} + \left(6 a^{3} + 5 a^{2} + 10 a + 4\right)\cdot 13^{3} + \left(5 a^{4} + 8 a^{3} + 12 a^{2} + 10 a + 8\right)\cdot 13^{4} + \left(9 a^{3} + 6 a^{2} + 6 a + 11\right)\cdot 13^{5} + \left(8 a^{4} + 10 a^{3} + 8 a + 4\right)\cdot 13^{6} + \left(9 a^{4} + 12 a^{3} + 8 a^{2} + a + 7\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 8 }$ $=$ $10 a^{4} + 3 a^{3} + 9 a + 6 + \left(3 a^{4} + 2 a^{3} + 6 a^{2} + 5 a + 9\right)\cdot 13 + \left(a^{4} + 6 a^{3} + 10 a^{2} + 10 a + 6\right)\cdot 13^{2} + \left(7 a^{4} + 6 a^{3} + 6 a^{2} + 7 a + 4\right)\cdot 13^{3} + \left(6 a^{4} + 4 a^{3} + 7 a^{2} + 12 a + 5\right)\cdot 13^{4} + \left(6 a^{4} + 9 a^{3} + a^{2} + 8 a + 10\right)\cdot 13^{5} + \left(11 a^{4} + 5 a^{3} + 9 a^{2} + 3 a + 10\right)\cdot 13^{6} + \left(6 a^{4} + 11 a^{2} + a + 3\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 9 }$ $=$ $2 a^{4} + 10 a^{2} + 9 + \left(2 a^{4} + a^{3} + 10 a^{2} + 7 a + 1\right)\cdot 13 + \left(9 a^{4} + 5 a^{3} + a^{2} + 7 a + 11\right)\cdot 13^{2} + \left(12 a^{4} + 11 a^{2} + 7 a + 6\right)\cdot 13^{3} + \left(8 a^{4} + 9 a^{3} + 9 a^{2} + 12 a + 10\right)\cdot 13^{4} + \left(a^{4} + 7 a^{3} + 5 a^{2} + 8 a + 2\right)\cdot 13^{5} + \left(11 a^{4} + 2 a^{3} + 3 a^{2} + a + 12\right)\cdot 13^{6} + \left(6 a^{3} + 5 a^{2} + 8 a + 7\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$ $r_{ 10 }$ $=$ $8 a^{4} + 5 a^{3} + 6 a^{2} + 3 a + 10 + \left(a^{4} + 11 a^{3} + 11 a^{2} + 3 a + 7\right)\cdot 13 + \left(10 a^{4} + 12 a^{3} + 7 a^{2} + 9 a + 11\right)\cdot 13^{2} + \left(6 a^{4} + 9 a^{3} + a^{2} + 3 a\right)\cdot 13^{3} + \left(a^{4} + 8 a^{3} + 12 a^{2} + 6 a + 10\right)\cdot 13^{4} + \left(a^{4} + 2 a^{3} + 8 a^{2} + 3 a + 8\right)\cdot 13^{5} + \left(3 a^{4} + 8 a^{3} + 8 a^{2} + 5 a + 4\right)\cdot 13^{6} + \left(7 a^{4} + 3 a^{3} + 6 a + 2\right)\cdot 13^{7} +O\left(13^{ 8 }\right)$

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

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

### Character values on conjugacy classes

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