Properties

Label 20.100...401.70.a.a
Dimension $20$
Group $S_7$
Conductor $1.002\times 10^{53}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $20$
Group: $S_7$
Conductor: \(100\!\cdots\!401\)\(\medspace = 199559^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.199559.1
Galois orbit size: $1$
Smallest permutation container: 70
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_7$
Projective stem field: Galois closure of 7.1.199559.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: \( x^{2} + 192x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 23 a + 18 + \left(190 a + 60\right)\cdot 193 + \left(181 a + 46\right)\cdot 193^{2} + \left(105 a + 42\right)\cdot 193^{3} + \left(13 a + 160\right)\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 57 a + 120 + \left(112 a + 18\right)\cdot 193 + \left(104 a + 55\right)\cdot 193^{2} + \left(125 a + 189\right)\cdot 193^{3} + \left(90 a + 151\right)\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 177 + 7\cdot 193 + 84\cdot 193^{2} + 133\cdot 193^{3} + 169\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 169 + 183\cdot 193 + 61\cdot 193^{2} + 130\cdot 193^{3} +O(193^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 170 a + 41 + \left(2 a + 34\right)\cdot 193 + \left(11 a + 38\right)\cdot 193^{2} + \left(87 a + 159\right)\cdot 193^{3} + \left(179 a + 67\right)\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 71 + 7\cdot 193 + 53\cdot 193^{2} + 100\cdot 193^{3} + 104\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 136 a + 177 + \left(80 a + 73\right)\cdot 193 + \left(88 a + 47\right)\cdot 193^{2} + \left(67 a + 17\right)\cdot 193^{3} + \left(102 a + 117\right)\cdot 193^{4} +O(193^{5})\) 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$$1$$()$$20$
$21$$2$$(1,2)$$0$
$105$$2$$(1,2)(3,4)(5,6)$$0$
$105$$2$$(1,2)(3,4)$$-4$
$70$$3$$(1,2,3)$$2$
$280$$3$$(1,2,3)(4,5,6)$$2$
$210$$4$$(1,2,3,4)$$0$
$630$$4$$(1,2,3,4)(5,6)$$0$
$504$$5$$(1,2,3,4,5)$$0$
$210$$6$$(1,2,3)(4,5)(6,7)$$2$
$420$$6$$(1,2,3)(4,5)$$0$
$840$$6$$(1,2,3,4,5,6)$$0$
$720$$7$$(1,2,3,4,5,6,7)$$-1$
$504$$10$$(1,2,3,4,5)(6,7)$$0$
$420$$12$$(1,2,3,4)(5,6,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.