Properties

Label 5.2e2_31_367.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 2^{2} \cdot 31 \cdot 367 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$45508= 2^{2} \cdot 31 \cdot 367 $
Artin number field: Splitting field of $f= x^{6} + x^{4} - x^{3} + 3 x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.2e2_31_367.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 307 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 307 }$: $ x^{2} + 306 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 27 a + 292 + \left(103 a + 278\right)\cdot 307 + \left(126 a + 206\right)\cdot 307^{2} + \left(117 a + 255\right)\cdot 307^{3} + \left(33 a + 190\right)\cdot 307^{4} +O\left(307^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 280 a + 12 + \left(203 a + 48\right)\cdot 307 + \left(180 a + 230\right)\cdot 307^{2} + \left(189 a + 246\right)\cdot 307^{3} + \left(273 a + 106\right)\cdot 307^{4} +O\left(307^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 186 a + 305 + \left(59 a + 58\right)\cdot 307 + \left(282 a + 47\right)\cdot 307^{2} + \left(209 a + 67\right)\cdot 307^{3} + \left(251 a + 17\right)\cdot 307^{4} +O\left(307^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 49 + 121\cdot 307 + 49\cdot 307^{2} + 180\cdot 307^{3} + 274\cdot 307^{4} +O\left(307^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 121 a + 184 + \left(247 a + 239\right)\cdot 307 + \left(24 a + 269\right)\cdot 307^{2} + \left(97 a + 301\right)\cdot 307^{3} + \left(55 a + 58\right)\cdot 307^{4} +O\left(307^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 79 + 174\cdot 307 + 117\cdot 307^{2} + 176\cdot 307^{3} + 272\cdot 307^{4} +O\left(307^{ 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 value
$1$$1$$()$$5$
$15$$2$$(1,2)(3,4)(5,6)$$-1$
$15$$2$$(1,2)$$3$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$2$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$90$$4$$(1,2,3,4)$$1$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.