Properties

Label 5.7_13_601.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 7 \cdot 13 \cdot 601 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$54691= 7 \cdot 13 \cdot 601 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 2 x^{3} + 2 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.7_13_601.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 29 a + 35 + \left(33 a + 45\right)\cdot 67 + \left(a + 45\right)\cdot 67^{2} + \left(38 a + 41\right)\cdot 67^{3} + \left(60 a + 24\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 45 a + 20 + \left(12 a + 3\right)\cdot 67 + \left(9 a + 36\right)\cdot 67^{2} + \left(8 a + 28\right)\cdot 67^{3} + \left(39 a + 20\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 38 a + 17 + \left(33 a + 16\right)\cdot 67 + \left(65 a + 18\right)\cdot 67^{2} + \left(28 a + 58\right)\cdot 67^{3} + \left(6 a + 27\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 10 a + 12 + \left(53 a + 29\right)\cdot 67 + \left(18 a + 9\right)\cdot 67^{2} + \left(23 a + 40\right)\cdot 67^{3} + 57\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 57 a + 52 + \left(13 a + 30\right)\cdot 67 + \left(48 a + 31\right)\cdot 67^{2} + \left(43 a + 47\right)\cdot 67^{3} + \left(66 a + 35\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 22 a + 66 + \left(54 a + 8\right)\cdot 67 + \left(57 a + 60\right)\cdot 67^{2} + \left(58 a + 51\right)\cdot 67^{3} + \left(27 a + 34\right)\cdot 67^{4} +O\left(67^{ 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.