Properties

Label 5.37253.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 37253 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$37253 $
Artin number field: Splitting field of $f= x^{6} - x^{4} + x^{2} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Even
Determinant: 1.37253.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 108 + 99\cdot 113 + 100\cdot 113^{2} + 76\cdot 113^{3} + 17\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 89 a + 68 + \left(104 a + 42\right)\cdot 113 + \left(44 a + 85\right)\cdot 113^{2} + \left(89 a + 12\right)\cdot 113^{3} + \left(94 a + 68\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 a + 6 + \left(8 a + 81\right)\cdot 113 + \left(68 a + 67\right)\cdot 113^{2} + \left(23 a + 23\right)\cdot 113^{3} + \left(18 a + 99\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 64 + 97\cdot 113 + 34\cdot 113^{2} + 46\cdot 113^{3} + 17\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 71 + 17\cdot 113 + 2\cdot 113^{2} + 44\cdot 113^{3} + 69\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 22 + 48\cdot 113^{2} + 22\cdot 113^{3} + 67\cdot 113^{4} +O\left(113^{ 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.