Properties

Label 6.214607.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 214607 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$214607 $
Artin number field: Splitting field of $f= x^{7} - x^{6} + 2 x^{5} - 2 x^{4} + 2 x^{3} - 2 x^{2} - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.214607.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 24 a + 3 + \left(4 a + 14\right)\cdot 31 + \left(24 a + 15\right)\cdot 31^{2} + \left(18 a + 6\right)\cdot 31^{3} + \left(11 a + 30\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 6 a + 27 + \left(4 a + 27\right)\cdot 31 + \left(28 a + 27\right)\cdot 31^{2} + \left(24 a + 13\right)\cdot 31^{3} + \left(25 a + 4\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 22 a + 25 + \left(27 a + 12\right)\cdot 31 + \left(8 a + 12\right)\cdot 31^{2} + \left(12 a + 11\right)\cdot 31^{3} + \left(24 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 9 a + 7 + \left(3 a + 15\right)\cdot 31 + \left(22 a + 2\right)\cdot 31^{2} + \left(18 a + 27\right)\cdot 31^{3} + \left(6 a + 28\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 4 + 24\cdot 31 + 19\cdot 31^{2} + 9\cdot 31^{3} + 2\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 25 a + 8 + \left(26 a + 30\right)\cdot 31 + \left(2 a + 17\right)\cdot 31^{2} + \left(6 a + 4\right)\cdot 31^{3} + 5 a\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 7 a + 20 + \left(26 a + 30\right)\cdot 31 + \left(6 a + 27\right)\cdot 31^{2} + \left(12 a + 19\right)\cdot 31^{3} + \left(19 a + 3\right)\cdot 31^{4} +O\left(31^{ 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.