# Properties

 Label 1.2e2_11_23.10t1.1c4 Dimension 1 Group $C_{10}$ Conductor $2^{2} \cdot 11 \cdot 23$ Root number not computed Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $1$ Group: $C_{10}$ Conductor: $1012= 2^{2} \cdot 11 \cdot 23$ Artin number field: Splitting field of $f= x^{10} - 2 x^{9} - 122 x^{8} + 198 x^{7} + 5582 x^{6} - 6794 x^{5} - 118785 x^{4} + 94556 x^{3} + 1168426 x^{2} - 446344 x - 4240279$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_{10}$ Parity: Even Corresponding Dirichlet character: $$\chi_{1012}(367,\cdot)$$

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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