Properties

Label 20.200...009.70.a.a
Dimension $20$
Group $S_7$
Conductor $2.009\times 10^{48}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $20$
Group: $S_7$
Conductor: \(200\!\cdots\!009\)\(\medspace = 53^{12} \cdot 577^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 7.3.1620793.1
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 7.3.1620793.1

Defining polynomial

$f(x)$$=$ \( x^{7} - 3x^{6} + 3x^{5} + x^{4} - 6x^{3} + 5x^{2} - x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 199 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 199 }$: \( x^{2} + 193x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 117 a + 15 + \left(57 a + 144\right)\cdot 199 + \left(38 a + 42\right)\cdot 199^{2} + \left(118 a + 191\right)\cdot 199^{3} + \left(106 a + 6\right)\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 136 a + 83 + \left(156 a + 60\right)\cdot 199 + \left(118 a + 95\right)\cdot 199^{2} + \left(119 a + 14\right)\cdot 199^{3} + \left(52 a + 162\right)\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 82 a + 120 + \left(141 a + 173\right)\cdot 199 + \left(160 a + 15\right)\cdot 199^{2} + \left(80 a + 66\right)\cdot 199^{3} + \left(92 a + 130\right)\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 112 a + 113 + \left(146 a + 179\right)\cdot 199 + \left(7 a + 169\right)\cdot 199^{2} + \left(34 a + 137\right)\cdot 199^{3} + \left(184 a + 93\right)\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 87 a + 188 + \left(52 a + 150\right)\cdot 199 + \left(191 a + 69\right)\cdot 199^{2} + \left(164 a + 135\right)\cdot 199^{3} + \left(14 a + 169\right)\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 177 + 18\cdot 199 + 149\cdot 199^{2} + 35\cdot 199^{3} + 74\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 63 a + 103 + \left(42 a + 68\right)\cdot 199 + \left(80 a + 54\right)\cdot 199^{2} + \left(79 a + 16\right)\cdot 199^{3} + \left(146 a + 159\right)\cdot 199^{4} +O(199^{5})\) 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.