Properties

Label 20.186...049.70.a.a
Dimension $20$
Group $S_7$
Conductor $1.865\times 10^{54}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $20$
Group: $S_7$
Conductor: \(186\!\cdots\!049\)\(\medspace = 101^{10} \cdot 2647^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.1.267347.1
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 7.1.267347.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 179 }$: \( x^{2} + 172x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 50 a + 108 + \left(151 a + 150\right)\cdot 179 + \left(84 a + 126\right)\cdot 179^{2} + \left(46 a + 60\right)\cdot 179^{3} + \left(118 a + 165\right)\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 157 a + 29 + \left(160 a + 35\right)\cdot 179 + \left(20 a + 160\right)\cdot 179^{2} + \left(97 a + 167\right)\cdot 179^{3} + \left(166 a + 160\right)\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 157 a + 144 + \left(44 a + 35\right)\cdot 179 + \left(138 a + 126\right)\cdot 179^{2} + \left(131 a + 107\right)\cdot 179^{3} + \left(82 a + 170\right)\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 22 a + 54 + \left(18 a + 109\right)\cdot 179 + \left(158 a + 145\right)\cdot 179^{2} + \left(81 a + 110\right)\cdot 179^{3} + \left(12 a + 155\right)\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 129 a + 100 + \left(27 a + 85\right)\cdot 179 + \left(94 a + 32\right)\cdot 179^{2} + \left(132 a + 122\right)\cdot 179^{3} + \left(60 a + 51\right)\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 22 a + 169 + \left(134 a + 13\right)\cdot 179 + \left(40 a + 154\right)\cdot 179^{2} + \left(47 a + 175\right)\cdot 179^{3} + \left(96 a + 80\right)\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 112 + 106\cdot 179 + 149\cdot 179^{2} + 149\cdot 179^{3} + 109\cdot 179^{4} +O(179^{5})\) 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.