Properties

Label 5.7_5867.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 7 \cdot 5867 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$41069= 7 \cdot 5867 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{4} + 2 x^{2} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Even
Determinant: 1.7_5867.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 20\cdot 37 + 15\cdot 37^{2} + 35\cdot 37^{3} + 19\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 13 a + 6 + \left(11 a + 2\right)\cdot 37 + \left(5 a + 23\right)\cdot 37^{2} + \left(3 a + 8\right)\cdot 37^{3} + \left(12 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 9 a + 26 + \left(28 a + 23\right)\cdot 37 + \left(11 a + 26\right)\cdot 37^{2} + \left(9 a + 30\right)\cdot 37^{3} + \left(22 a + 21\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 31 + 13\cdot 37 + 4\cdot 37^{2} + 37^{3} + 26\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 28 a + 25 + \left(8 a + 16\right)\cdot 37 + \left(25 a + 8\right)\cdot 37^{2} + \left(27 a + 19\right)\cdot 37^{3} + \left(14 a + 27\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 24 a + 21 + \left(25 a + 34\right)\cdot 37 + \left(31 a + 32\right)\cdot 37^{2} + \left(33 a + 15\right)\cdot 37^{3} + \left(24 a + 30\right)\cdot 37^{4} +O\left(37^{ 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.