Properties

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

Related objects

Learn more about

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

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

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $10$
$15$ $2$ $(1,2)(3,4)(5,6)$ $-2$
$15$ $2$ $(1,2)$ $2$
$45$ $2$ $(1,2)(3,4)$ $-2$
$40$ $3$ $(1,2,3)(4,5,6)$ $1$
$40$ $3$ $(1,2,3)$ $1$
$90$ $4$ $(1,2,3,4)(5,6)$ $0$
$90$ $4$ $(1,2,3,4)$ $0$
$144$ $5$ $(1,2,3,4,5)$ $0$
$120$ $6$ $(1,2,3,4,5,6)$ $1$
$120$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.