Properties

Label 5.131_383.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 131 \cdot 383 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$50173= 131 \cdot 383 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} + 2 x^{3} - 4 x^{2} + 3 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Even
Determinant: 1.131_383.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 5 + 91\cdot 97 + 75\cdot 97^{2} + 36\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 66 a + 3 + \left(28 a + 43\right)\cdot 97 + \left(16 a + 75\right)\cdot 97^{2} + \left(70 a + 66\right)\cdot 97^{3} + \left(26 a + 34\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 31 + 12\cdot 97 + 72\cdot 97^{2} + 47\cdot 97^{3} + 92\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 87 a + 31 a\cdot 97 + \left(a + 66\right)\cdot 97^{2} + \left(54 a + 49\right)\cdot 97^{3} + \left(71 a + 59\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 10 a + 87 + \left(65 a + 41\right)\cdot 97 + \left(95 a + 35\right)\cdot 97^{2} + \left(42 a + 5\right)\cdot 97^{3} + \left(25 a + 77\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 31 a + 69 + \left(68 a + 5\right)\cdot 97 + \left(80 a + 63\right)\cdot 97^{2} + \left(26 a + 23\right)\cdot 97^{3} + \left(70 a + 88\right)\cdot 97^{4} +O\left(97^{ 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.