Properties

Label 28.638...504.56.a.a
Dimension $28$
Group $A_8$
Conductor $6.385\times 10^{208}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $28$
Group: $A_8$
Conductor: \(638\!\cdots\!504\)\(\medspace = 2^{78} \cdot 52706761^{24}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.0.5620020392178081715128903113480438691650659827318784.1
Galois orbit size: $1$
Smallest permutation container: 56
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: Galois closure of 8.0.5620020392178081715128903113480438691650659827318784.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825252 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 239 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 239 }$: \( x^{2} + 237x + 7 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 107 a + 177 + \left(226 a + 157\right)\cdot 239 + \left(192 a + 95\right)\cdot 239^{2} + \left(110 a + 152\right)\cdot 239^{3} + \left(97 a + 158\right)\cdot 239^{4} + \left(209 a + 215\right)\cdot 239^{5} + \left(117 a + 49\right)\cdot 239^{6} + \left(52 a + 138\right)\cdot 239^{7} + \left(214 a + 132\right)\cdot 239^{8} + \left(170 a + 113\right)\cdot 239^{9} +O(239^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 175 + 239 + 225\cdot 239^{2} + 204\cdot 239^{3} + 136\cdot 239^{4} + 199\cdot 239^{5} + 175\cdot 239^{6} + 225\cdot 239^{7} + 83\cdot 239^{8} + 206\cdot 239^{9} +O(239^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 225 a + 228 + \left(189 a + 88\right)\cdot 239 + \left(177 a + 10\right)\cdot 239^{2} + \left(166 a + 117\right)\cdot 239^{3} + \left(76 a + 8\right)\cdot 239^{4} + \left(225 a + 52\right)\cdot 239^{5} + \left(5 a + 21\right)\cdot 239^{6} + \left(194 a + 2\right)\cdot 239^{7} + \left(133 a + 67\right)\cdot 239^{8} + \left(224 a + 138\right)\cdot 239^{9} +O(239^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 139 + 147\cdot 239 + 124\cdot 239^{2} + 79\cdot 239^{3} + 47\cdot 239^{4} + 4\cdot 239^{5} + 160\cdot 239^{6} + 36\cdot 239^{7} + 129\cdot 239^{8} + 124\cdot 239^{9} +O(239^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 132 a + 152 + \left(12 a + 25\right)\cdot 239 + \left(46 a + 16\right)\cdot 239^{2} + \left(128 a + 181\right)\cdot 239^{3} + \left(141 a + 3\right)\cdot 239^{4} + \left(29 a + 59\right)\cdot 239^{5} + \left(121 a + 76\right)\cdot 239^{6} + \left(186 a + 125\right)\cdot 239^{7} + \left(24 a + 30\right)\cdot 239^{8} + \left(68 a + 2\right)\cdot 239^{9} +O(239^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 14 a + 200 + \left(49 a + 4\right)\cdot 239 + \left(61 a + 176\right)\cdot 239^{2} + \left(72 a + 33\right)\cdot 239^{3} + \left(162 a + 234\right)\cdot 239^{4} + \left(13 a + 186\right)\cdot 239^{5} + \left(233 a + 46\right)\cdot 239^{6} + \left(44 a + 145\right)\cdot 239^{7} + \left(105 a + 140\right)\cdot 239^{8} + \left(14 a + 214\right)\cdot 239^{9} +O(239^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 68 + 4\cdot 239 + 82\cdot 239^{2} + 17\cdot 239^{3} + 234\cdot 239^{4} + 4\cdot 239^{5} + 183\cdot 239^{6} + 158\cdot 239^{7} + 200\cdot 239^{8} + 213\cdot 239^{9} +O(239^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 56 + 47\cdot 239 + 226\cdot 239^{2} + 169\cdot 239^{3} + 132\cdot 239^{4} + 233\cdot 239^{5} + 3\cdot 239^{6} + 124\cdot 239^{7} + 171\cdot 239^{8} + 181\cdot 239^{9} +O(239^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,2)(3,4,5,6,7,8)$
$(1,2,3)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$28$
$105$$2$$(1,2)(3,4)(5,6)(7,8)$$-4$
$210$$2$$(1,2)(3,4)$$4$
$112$$3$$(1,2,3)$$1$
$1120$$3$$(1,2,3)(4,5,6)$$1$
$1260$$4$$(1,2,3,4)(5,6,7,8)$$0$
$2520$$4$$(1,2,3,4)(5,6)$$0$
$1344$$5$$(1,2,3,4,5)$$-2$
$1680$$6$$(1,2,3)(4,5)(6,7)$$1$
$3360$$6$$(1,2,3,4,5,6)(7,8)$$-1$
$2880$$7$$(1,2,3,4,5,6,7)$$0$
$2880$$7$$(1,3,4,5,6,7,2)$$0$
$1344$$15$$(1,2,3,4,5)(6,7,8)$$1$
$1344$$15$$(1,3,4,5,2)(6,7,8)$$1$

The blue line marks the conjugacy class containing complex conjugation.