Properties

Label 6.17_11863.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 17 \cdot 11863 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$201671= 17 \cdot 11863 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + 2 x^{5} - 2 x^{4} + 2 x^{3} - 2 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.17_11863.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: $ x^{2} + 274 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 168 a + 90 + \left(208 a + 2\right)\cdot 277 + \left(236 a + 96\right)\cdot 277^{2} + \left(152 a + 64\right)\cdot 277^{3} + \left(129 a + 87\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 177 a + 107 + \left(86 a + 169\right)\cdot 277 + \left(32 a + 270\right)\cdot 277^{2} + \left(165 a + 183\right)\cdot 277^{3} + \left(63 a + 168\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 167 a + 114 + \left(265 a + 45\right)\cdot 277 + \left(177 a + 249\right)\cdot 277^{2} + \left(125 a + 27\right)\cdot 277^{3} + \left(57 a + 75\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 100 a + 84 + \left(190 a + 252\right)\cdot 277 + \left(244 a + 3\right)\cdot 277^{2} + \left(111 a + 93\right)\cdot 277^{3} + \left(213 a + 194\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 59 + 57\cdot 277 + 204\cdot 277^{2} + 225\cdot 277^{3} + 137\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 109 a + 40 + \left(68 a + 183\right)\cdot 277 + \left(40 a + 43\right)\cdot 277^{2} + \left(124 a + 9\right)\cdot 277^{3} + \left(147 a + 46\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 110 a + 61 + \left(11 a + 121\right)\cdot 277 + \left(99 a + 240\right)\cdot 277^{2} + \left(151 a + 226\right)\cdot 277^{3} + \left(219 a + 121\right)\cdot 277^{4} +O\left(277^{ 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.