Properties

Label 6.252071.7t7.a.a
Dimension $6$
Group $S_7$
Conductor $252071$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_7$
Conductor: \(252071\)\(\medspace = 83 \cdot 3037 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.252071.1
Galois orbit size: $1$
Smallest permutation container: $S_7$
Parity: odd
Determinant: 1.252071.2t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.252071.1

Defining polynomial

$f(x)$$=$ \( x^{7} - x^{6} + 2x^{4} - x^{3} - 2x^{2} + x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: \( x^{2} + 166x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 144 + 16\cdot 167 + 146\cdot 167^{2} + 130\cdot 167^{3} + 61\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 58 + 84\cdot 167 + 31\cdot 167^{2} + 68\cdot 167^{3} + 89\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 83 a + 16 + \left(72 a + 80\right)\cdot 167 + \left(71 a + 93\right)\cdot 167^{2} + \left(14 a + 104\right)\cdot 167^{3} + \left(48 a + 10\right)\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 35 a + 41 + \left(126 a + 134\right)\cdot 167 + \left(62 a + 7\right)\cdot 167^{2} + \left(129 a + 10\right)\cdot 167^{3} + \left(41 a + 148\right)\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 84 a + 99 + \left(94 a + 69\right)\cdot 167 + \left(95 a + 92\right)\cdot 167^{2} + \left(152 a + 47\right)\cdot 167^{3} + \left(118 a + 44\right)\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 132 a + 76 + \left(40 a + 58\right)\cdot 167 + \left(104 a + 111\right)\cdot 167^{2} + \left(37 a + 76\right)\cdot 167^{3} + \left(125 a + 60\right)\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 68 + 57\cdot 167 + 18\cdot 167^{2} + 63\cdot 167^{3} + 86\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display

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.