Properties

Label 6.193607.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 193607 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$193607 $
Artin number field: Splitting field of $f= x^{7} - x^{4} - x^{3} + x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.193607.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
$r_{ 1 }$ $=$ $ 69 a + 69 + \left(67 a + 48\right)\cdot 71 + \left(13 a + 41\right)\cdot 71^{2} + \left(65 a + 56\right)\cdot 71^{3} + \left(32 a + 1\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 54 + 34\cdot 71 + 41\cdot 71^{2} + 48\cdot 71^{3} + 15\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 68 a + 66 + \left(23 a + 36\right)\cdot 71 + \left(14 a + 45\right)\cdot 71^{2} + \left(22 a + 69\right)\cdot 71^{3} + \left(17 a + 21\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 2 a + 65 + \left(3 a + 44\right)\cdot 71 + \left(57 a + 1\right)\cdot 71^{2} + \left(5 a + 31\right)\cdot 71^{3} + \left(38 a + 2\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 12 + 15\cdot 71 + 56\cdot 71^{2} + 27\cdot 71^{3} + 10\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 3 a + 60 + \left(47 a + 16\right)\cdot 71 + \left(56 a + 50\right)\cdot 71^{2} + \left(48 a + 28\right)\cdot 71^{3} + \left(53 a + 34\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 29 + 15\cdot 71 + 47\cdot 71^{2} + 21\cdot 71^{3} + 55\cdot 71^{4} +O\left(71^{ 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.