Properties

Label 5.41_1103.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 41 \cdot 1103 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$45223= 41 \cdot 1103 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - x^{4} - x^{3} + 2 x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.41_1103.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 18\cdot 19 + 4\cdot 19^{2} + 17\cdot 19^{3} + 11\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 a + 8 a\cdot 19 + \left(8 a + 16\right)\cdot 19^{2} + \left(5 a + 8\right)\cdot 19^{3} + \left(14 a + 7\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 17 a + 2 + \left(10 a + 6\right)\cdot 19 + \left(10 a + 16\right)\cdot 19^{2} + \left(13 a + 5\right)\cdot 19^{3} + \left(4 a + 16\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 3 a + 17 + \left(16 a + 8\right)\cdot 19 + \left(14 a + 4\right)\cdot 19^{2} + \left(9 a + 11\right)\cdot 19^{3} + 8\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 17 + 19 + 12\cdot 19^{2} + 7\cdot 19^{3} + 13\cdot 19^{4} +O\left(19^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 16 a + 1 + \left(2 a + 3\right)\cdot 19 + \left(4 a + 3\right)\cdot 19^{2} + \left(9 a + 6\right)\cdot 19^{3} + \left(18 a + 18\right)\cdot 19^{4} +O\left(19^{ 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.