Properties

Label 5.73_653.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 73 \cdot 653 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$47669= 73 \cdot 653 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} - x^{2} - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Even
Determinant: 1.73_653.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 397 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 397 }$: $ x^{2} + 392 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 282 + 305\cdot 397 + 79\cdot 397^{2} + 80\cdot 397^{3} + 121\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 301 + 164\cdot 397 + 135\cdot 397^{2} + 362\cdot 397^{3} + 56\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 131 a + 373 + \left(208 a + 189\right)\cdot 397 + \left(321 a + 333\right)\cdot 397^{2} + 3 a\cdot 397^{3} + \left(49 a + 79\right)\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 364 a + 282 + \left(96 a + 51\right)\cdot 397 + \left(71 a + 119\right)\cdot 397^{2} + \left(311 a + 179\right)\cdot 397^{3} + \left(127 a + 341\right)\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 266 a + 234 + \left(188 a + 306\right)\cdot 397 + \left(75 a + 144\right)\cdot 397^{2} + \left(393 a + 95\right)\cdot 397^{3} + \left(347 a + 320\right)\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 33 a + 117 + \left(300 a + 172\right)\cdot 397 + \left(325 a + 378\right)\cdot 397^{2} + \left(85 a + 75\right)\cdot 397^{3} + \left(269 a + 272\right)\cdot 397^{4} +O\left(397^{ 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.