Properties

Label 45.782...528.336.a.b
Dimension $45$
Group $A_8$
Conductor $7.827\times 10^{166}$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $45$
Group: $A_8$
Conductor: \(782\!\cdots\!528\)\(\medspace = 2^{100} \cdot 3217^{39}\)
Artin stem field: Galois closure of 8.0.72641749645773438449680384.1
Galois orbit size: $2$
Smallest permutation container: 336
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: Galois closure of 8.0.72641749645773438449680384.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 3217 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 8.

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

Roots:
$r_{ 1 }$ $=$ \( 16 + 153\cdot 179 + 69\cdot 179^{2} + 79\cdot 179^{3} + 94\cdot 179^{4} + 38\cdot 179^{5} + 161\cdot 179^{6} + 108\cdot 179^{7} +O(179^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 137 + 154\cdot 179 + 67\cdot 179^{2} + 54\cdot 179^{3} + 108\cdot 179^{4} + 156\cdot 179^{5} + 26\cdot 179^{6} + 172\cdot 179^{7} +O(179^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 122 + 83\cdot 179 + 163\cdot 179^{2} + 12\cdot 179^{3} + 177\cdot 179^{4} + 156\cdot 179^{5} + 26\cdot 179^{6} + 39\cdot 179^{7} +O(179^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 137 a + 138 + \left(176 a + 107\right)\cdot 179 + \left(7 a + 119\right)\cdot 179^{2} + \left(113 a + 141\right)\cdot 179^{3} + 70\cdot 179^{4} + \left(90 a + 109\right)\cdot 179^{5} + \left(88 a + 127\right)\cdot 179^{6} + \left(20 a + 113\right)\cdot 179^{7} +O(179^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 46 a + 71 + \left(31 a + 178\right)\cdot 179 + \left(54 a + 127\right)\cdot 179^{2} + \left(172 a + 112\right)\cdot 179^{3} + \left(102 a + 56\right)\cdot 179^{4} + \left(113 a + 176\right)\cdot 179^{5} + \left(78 a + 108\right)\cdot 179^{6} + \left(30 a + 100\right)\cdot 179^{7} +O(179^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 178 + 89\cdot 179 + 50\cdot 179^{2} + 94\cdot 179^{3} + 178\cdot 179^{4} + 81\cdot 179^{5} + 135\cdot 179^{6} + 135\cdot 179^{7} +O(179^{8})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 42 a + 23 + \left(2 a + 134\right)\cdot 179 + \left(171 a + 177\right)\cdot 179^{2} + \left(65 a + 29\right)\cdot 179^{3} + \left(178 a + 141\right)\cdot 179^{4} + \left(88 a + 22\right)\cdot 179^{5} + \left(90 a + 120\right)\cdot 179^{6} + \left(158 a + 168\right)\cdot 179^{7} +O(179^{8})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 133 a + 35 + \left(147 a + 172\right)\cdot 179 + \left(124 a + 117\right)\cdot 179^{2} + \left(6 a + 11\right)\cdot 179^{3} + \left(76 a + 68\right)\cdot 179^{4} + \left(65 a + 152\right)\cdot 179^{5} + \left(100 a + 8\right)\cdot 179^{6} + \left(148 a + 56\right)\cdot 179^{7} +O(179^{8})\) 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$$1$$()$$45$
$105$$2$$(1,2)(3,4)(5,6)(7,8)$$-3$
$210$$2$$(1,2)(3,4)$$-3$
$112$$3$$(1,2,3)$$0$
$1120$$3$$(1,2,3)(4,5,6)$$0$
$1260$$4$$(1,2,3,4)(5,6,7,8)$$1$
$2520$$4$$(1,2,3,4)(5,6)$$1$
$1344$$5$$(1,2,3,4,5)$$0$
$1680$$6$$(1,2,3)(4,5)(6,7)$$0$
$3360$$6$$(1,2,3,4,5,6)(7,8)$$0$
$2880$$7$$(1,2,3,4,5,6,7)$$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
$2880$$7$$(1,3,4,5,6,7,2)$$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
$1344$$15$$(1,2,3,4,5)(6,7,8)$$0$
$1344$$15$$(1,3,4,5,2)(6,7,8)$$0$

The blue line marks the conjugacy class containing complex conjugation.