# Properties

 Label 10.2e6_3e4_17e6_23e4.30t176.1c1 Dimension 10 Group $S_6$ Conductor $2^{6} \cdot 3^{4} \cdot 17^{6} \cdot 23^{4}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $35016268618806336= 2^{6} \cdot 3^{4} \cdot 17^{6} \cdot 23^{4}$ Artin number field: Splitting field of $f= x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 2 x^{2} + 3 x - 3$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 30T176 Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 401 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 401 }$: $x^{2} + 396 x + 3$
Roots:
 $r_{ 1 }$ $=$ $195 a + 103 + \left(352 a + 27\right)\cdot 401 + \left(127 a + 120\right)\cdot 401^{2} + \left(73 a + 211\right)\cdot 401^{3} + \left(331 a + 187\right)\cdot 401^{4} +O\left(401^{ 5 }\right)$ $r_{ 2 }$ $=$ $349 + 218\cdot 401 + 191\cdot 401^{2} + 127\cdot 401^{3} + 374\cdot 401^{4} +O\left(401^{ 5 }\right)$ $r_{ 3 }$ $=$ $206 a + 276 + \left(48 a + 391\right)\cdot 401 + \left(273 a + 5\right)\cdot 401^{2} + \left(327 a + 49\right)\cdot 401^{3} + \left(69 a + 166\right)\cdot 401^{4} +O\left(401^{ 5 }\right)$ $r_{ 4 }$ $=$ $371 + 317\cdot 401 + 306\cdot 401^{2} + 34\cdot 401^{3} + 14\cdot 401^{4} +O\left(401^{ 5 }\right)$ $r_{ 5 }$ $=$ $62 a + 98 + \left(377 a + 214\right)\cdot 401 + \left(139 a + 328\right)\cdot 401^{2} + \left(163 a + 51\right)\cdot 401^{3} + \left(256 a + 72\right)\cdot 401^{4} +O\left(401^{ 5 }\right)$ $r_{ 6 }$ $=$ $339 a + 7 + \left(23 a + 33\right)\cdot 401 + \left(261 a + 250\right)\cdot 401^{2} + \left(237 a + 327\right)\cdot 401^{3} + \left(144 a + 388\right)\cdot 401^{4} +O\left(401^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,2)$ $(1,2,3,4,5,6)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $10$ $15$ $2$ $(1,2)(3,4)(5,6)$ $-2$ $15$ $2$ $(1,2)$ $2$ $45$ $2$ $(1,2)(3,4)$ $-2$ $40$ $3$ $(1,2,3)(4,5,6)$ $1$ $40$ $3$ $(1,2,3)$ $1$ $90$ $4$ $(1,2,3,4)(5,6)$ $0$ $90$ $4$ $(1,2,3,4)$ $0$ $144$ $5$ $(1,2,3,4,5)$ $0$ $120$ $6$ $(1,2,3,4,5,6)$ $1$ $120$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.