Properties

Label 45.193...312.336.a.b
Dimension $45$
Group $A_8$
Conductor $1.935\times 10^{340}$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $45$
Group: $A_8$
Conductor: \(193\!\cdots\!312\)\(\medspace = 2^{130} \cdot 23^{39} \cdot 43^{39} \cdot 137^{39} \cdot 389^{39}\)
Artin stem field: Galois closure of 8.0.5620030628480917722529796097626934820084447048892416.1
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 8.0.5620030628480917722529796097626934820084447048892416.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825316 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 349 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 349 }$: \( x^{2} + 348x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 109\cdot 349 + 166\cdot 349^{2} + 349^{3} + 28\cdot 349^{4} + 30\cdot 349^{5} + 328\cdot 349^{6} + 85\cdot 349^{7} + 132\cdot 349^{8} + 205\cdot 349^{9} +O(349^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 214 a + 175 + \left(44 a + 98\right)\cdot 349 + \left(345 a + 74\right)\cdot 349^{2} + \left(65 a + 112\right)\cdot 349^{3} + \left(96 a + 233\right)\cdot 349^{4} + \left(185 a + 26\right)\cdot 349^{5} + \left(283 a + 37\right)\cdot 349^{6} + \left(206 a + 292\right)\cdot 349^{7} + \left(96 a + 104\right)\cdot 349^{8} + \left(231 a + 139\right)\cdot 349^{9} +O(349^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 146 + 291\cdot 349 + 134\cdot 349^{2} + 186\cdot 349^{3} + 287\cdot 349^{5} + 120\cdot 349^{6} + 239\cdot 349^{7} + 217\cdot 349^{8} + 191\cdot 349^{9} +O(349^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 27 + 105\cdot 349 + 92\cdot 349^{2} + 197\cdot 349^{3} + 158\cdot 349^{4} + 327\cdot 349^{5} + 14\cdot 349^{6} + 40\cdot 349^{7} + 14\cdot 349^{8} + 170\cdot 349^{9} +O(349^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 309 a + 80 + \left(208 a + 170\right)\cdot 349 + \left(142 a + 6\right)\cdot 349^{2} + \left(239 a + 149\right)\cdot 349^{3} + \left(184 a + 176\right)\cdot 349^{4} + \left(127 a + 71\right)\cdot 349^{5} + \left(305 a + 309\right)\cdot 349^{6} + \left(250 a + 197\right)\cdot 349^{7} + \left(28 a + 287\right)\cdot 349^{8} + \left(a + 159\right)\cdot 349^{9} +O(349^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 40 a + 40 + \left(140 a + 70\right)\cdot 349 + \left(206 a + 289\right)\cdot 349^{2} + \left(109 a + 245\right)\cdot 349^{3} + \left(164 a + 121\right)\cdot 349^{4} + \left(221 a + 14\right)\cdot 349^{5} + \left(43 a + 138\right)\cdot 349^{6} + \left(98 a + 143\right)\cdot 349^{7} + \left(320 a + 65\right)\cdot 349^{8} + \left(347 a + 132\right)\cdot 349^{9} +O(349^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 190 + 273\cdot 349 + 257\cdot 349^{2} + 321\cdot 349^{3} + 64\cdot 349^{4} + 174\cdot 349^{5} + 312\cdot 349^{6} + 181\cdot 349^{7} + 230\cdot 349^{8} + 123\cdot 349^{9} +O(349^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 135 a + 40 + \left(304 a + 278\right)\cdot 349 + \left(3 a + 25\right)\cdot 349^{2} + \left(283 a + 182\right)\cdot 349^{3} + \left(252 a + 263\right)\cdot 349^{4} + \left(163 a + 115\right)\cdot 349^{5} + \left(65 a + 135\right)\cdot 349^{6} + \left(142 a + 215\right)\cdot 349^{7} + \left(252 a + 343\right)\cdot 349^{8} + \left(117 a + 273\right)\cdot 349^{9} +O(349^{10})\) 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$$1$$()$$45$
$105$$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.