# Properties

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

# Related objects

## 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

 $f(x)$ $=$ $$x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825316$$ x^8 - 112*x^6 - 896*x^5 - 3360*x^4 - 7168*x^3 - 8960*x^2 - 6144*x + 210825316 .

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 349 }$: $$x^{2} + 348x + 2$$

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

## 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

 Size Order Action 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.