Properties

Label 6.289987.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 289987 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$289987 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + 2 x^{5} - 2 x^{4} + 2 x^{3} - x^{2} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.289987.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 491 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 491 }$: $ x^{2} + 487 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 292 a + 4 + \left(91 a + 406\right)\cdot 491 + \left(249 a + 160\right)\cdot 491^{2} + \left(305 a + 160\right)\cdot 491^{3} + \left(243 a + 302\right)\cdot 491^{4} +O\left(491^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 199 a + 190 + \left(399 a + 480\right)\cdot 491 + \left(241 a + 83\right)\cdot 491^{2} + \left(185 a + 151\right)\cdot 491^{3} + \left(247 a + 480\right)\cdot 491^{4} +O\left(491^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 460 a + 14 + \left(391 a + 208\right)\cdot 491 + \left(358 a + 435\right)\cdot 491^{2} + \left(93 a + 232\right)\cdot 491^{3} + \left(389 a + 22\right)\cdot 491^{4} +O\left(491^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 12 a + 130 + \left(293 a + 191\right)\cdot 491 + \left(29 a + 486\right)\cdot 491^{2} + \left(120 a + 31\right)\cdot 491^{3} + \left(249 a + 33\right)\cdot 491^{4} +O\left(491^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 479 a + 178 + \left(197 a + 369\right)\cdot 491 + \left(461 a + 311\right)\cdot 491^{2} + \left(370 a + 482\right)\cdot 491^{3} + \left(241 a + 418\right)\cdot 491^{4} +O\left(491^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 86 + 466\cdot 491 + 479\cdot 491^{2} + 164\cdot 491^{3} + 203\cdot 491^{4} +O\left(491^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 31 a + 381 + \left(99 a + 333\right)\cdot 491 + \left(132 a + 5\right)\cdot 491^{2} + \left(397 a + 249\right)\cdot 491^{3} + \left(101 a + 12\right)\cdot 491^{4} +O\left(491^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation
$(1,2,3,4,5,6,7)$
$(1,2)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 7 }$ Character value
$1$$1$$()$$6$
$21$$2$$(1,2)$$4$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$2$
$70$$3$$(1,2,3)$$3$
$280$$3$$(1,2,3)(4,5,6)$$0$
$210$$4$$(1,2,3,4)$$2$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$1$
$210$$6$$(1,2,3)(4,5)(6,7)$$-1$
$420$$6$$(1,2,3)(4,5)$$1$
$840$$6$$(1,2,3,4,5,6)$$0$
$720$$7$$(1,2,3,4,5,6,7)$$-1$
$504$$10$$(1,2,3,4,5)(6,7)$$-1$
$420$$12$$(1,2,3,4)(5,6,7)$$-1$
The blue line marks the conjugacy class containing complex conjugation.