# Properties

 Label 1.2e2_11.10t1.1c2 Dimension 1 Group $C_{10}$ Conductor $2^{2} \cdot 11$ Root number not computed Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $1$ Group: $C_{10}$ Conductor: $44= 2^{2} \cdot 11$ Artin number field: Splitting field of $f= x^{10} + 9 x^{8} + 28 x^{6} + 35 x^{4} + 15 x^{2} + 1$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_{10}$ Parity: Odd Corresponding Dirichlet character: $$\chi_{44}(31,\cdot)$$

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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