Properties

Label 5.5_7247.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 5 \cdot 7247 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$36235= 5 \cdot 7247 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.5_7247.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 22 a + 2 + 16 a\cdot 31 + \left(5 a + 6\right)\cdot 31^{2} + \left(14 a + 10\right)\cdot 31^{3} + \left(6 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 22 + 19\cdot 31 + 29\cdot 31^{2} + 14\cdot 31^{3} + 3\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 5 a + \left(2 a + 23\right)\cdot 31 + 5 a\cdot 31^{2} + \left(22 a + 12\right)\cdot 31^{3} + \left(29 a + 13\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 26 a + 10 + \left(28 a + 22\right)\cdot 31 + \left(25 a + 8\right)\cdot 31^{2} + \left(8 a + 20\right)\cdot 31^{3} + \left(a + 19\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 14 + 16\cdot 31 + 16\cdot 31^{2} + 2\cdot 31^{3} + 21\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 9 a + 15 + \left(14 a + 11\right)\cdot 31 + 25 a\cdot 31^{2} + \left(16 a + 2\right)\cdot 31^{3} + \left(24 a + 17\right)\cdot 31^{4} +O\left(31^{ 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.