Properties

Label 6.313_1307.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 313 \cdot 1307 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$409091= 313 \cdot 1307 $
Artin number field: Splitting field of $f= x^{7} - x^{5} + 2 x^{3} - 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.313_1307.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 353 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 353 }$: $ x^{2} + 348 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 232 a + 227 + \left(27 a + 2\right)\cdot 353 + \left(298 a + 81\right)\cdot 353^{2} + \left(349 a + 121\right)\cdot 353^{3} + \left(152 a + 338\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 118 a + 151 + \left(107 a + 162\right)\cdot 353 + \left(302 a + 132\right)\cdot 353^{2} + \left(134 a + 105\right)\cdot 353^{3} + \left(84 a + 279\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 243 + 159\cdot 353 + 14\cdot 353^{2} + 247\cdot 353^{3} + 192\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 121 a + 328 + \left(325 a + 261\right)\cdot 353 + \left(54 a + 131\right)\cdot 353^{2} + \left(3 a + 160\right)\cdot 353^{3} + \left(200 a + 47\right)\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 130 + 229\cdot 353 + 177\cdot 353^{2} + 315\cdot 353^{3} + 193\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 298 + 14\cdot 353 + 44\cdot 353^{2} + 338\cdot 353^{3} + 146\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 235 a + 35 + \left(245 a + 228\right)\cdot 353 + \left(50 a + 124\right)\cdot 353^{2} + \left(218 a + 124\right)\cdot 353^{3} + \left(268 a + 213\right)\cdot 353^{4} +O\left(353^{ 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.