Properties

Label 5.2e4_47_61.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 2^{4} \cdot 47 \cdot 61 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$45872= 2^{4} \cdot 47 \cdot 61 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - x^{3} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.47_61.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 4 + 3\cdot 23 + 16\cdot 23^{2} + 18\cdot 23^{3} + 6\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 8 + 10\cdot 23 + 21\cdot 23^{2} + 6\cdot 23^{3} + 18\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ a + 10 + \left(4 a + 2\right)\cdot 23 + \left(21 a + 15\right)\cdot 23^{2} + \left(20 a + 4\right)\cdot 23^{3} + \left(4 a + 21\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 22 a + 12 + \left(18 a + 9\right)\cdot 23 + \left(a + 7\right)\cdot 23^{2} + \left(2 a + 2\right)\cdot 23^{3} + \left(18 a + 10\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 10 a + 8 + 3\cdot 23 + \left(14 a + 2\right)\cdot 23^{2} + \left(22 a + 14\right)\cdot 23^{3} + \left(13 a + 3\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 13 a + 5 + \left(22 a + 17\right)\cdot 23 + \left(8 a + 6\right)\cdot 23^{2} + 22\cdot 23^{3} + \left(9 a + 8\right)\cdot 23^{4} +O\left(23^{ 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.