Properties

Label 6.351587.7t7.1c1
Dimension 6
Group $S_7$
Conductor $ 351587 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$S_7$
Conductor:$351587 $
Artin number field: Splitting field of $f= x^{7} - 2 x^{5} - 2 x^{4} + 2 x^{3} + 3 x^{2} - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_7$
Parity: Odd
Determinant: 1.351587.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 4 a + 9 + \left(34 a + 4\right)\cdot 47 + \left(6 a + 38\right)\cdot 47^{2} + \left(29 a + 29\right)\cdot 47^{3} + \left(24 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 6 a + 32 + \left(32 a + 42\right)\cdot 47 + \left(20 a + 18\right)\cdot 47^{2} + \left(28 a + 33\right)\cdot 47^{3} + \left(20 a + 28\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 41 a + 44 + \left(14 a + 6\right)\cdot 47 + \left(26 a + 28\right)\cdot 47^{2} + \left(18 a + 22\right)\cdot 47^{3} + \left(26 a + 41\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 22 a + 8 + \left(46 a + 24\right)\cdot 47 + \left(32 a + 35\right)\cdot 47^{2} + \left(8 a + 3\right)\cdot 47^{3} + \left(40 a + 34\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 43 a + 17 + \left(12 a + 21\right)\cdot 47 + \left(40 a + 17\right)\cdot 47^{2} + \left(17 a + 34\right)\cdot 47^{3} + \left(22 a + 32\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 26 + 40\cdot 47 + 41\cdot 47^{2} + 28\cdot 47^{3} + 26\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 25 a + 5 + 47 + \left(14 a + 8\right)\cdot 47^{2} + \left(38 a + 35\right)\cdot 47^{3} + \left(6 a + 11\right)\cdot 47^{4} +O\left(47^{ 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.