# Properties

 Label 45.736...656.336.a.b Dimension $45$ Group $A_8$ Conductor $7.362\times 10^{275}$ Root number not computed Indicator $0$

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## Basic invariants

 Dimension: $45$ Group: $A_8$ Conductor: $$736\!\cdots\!656$$$$\medspace = 2^{150} \cdot 43^{39} \cdot 107^{39} \cdot 179^{39}$$ Artin stem field: Galois closure of 8.0.81803428774472904272307991671908472193024.1 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 8.0.81803428774472904272307991671908472193024.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823572$$ x^8 - 28*x^6 - 112*x^5 - 210*x^4 - 224*x^3 - 140*x^2 - 48*x + 823572 .

The roots of $f$ are computed in an extension of $\Q_{ 307 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 307 }$: $$x^{2} + 306x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$303 a + 119 + \left(115 a + 212\right)\cdot 307 + \left(238 a + 19\right)\cdot 307^{2} + \left(217 a + 39\right)\cdot 307^{3} + \left(104 a + 185\right)\cdot 307^{4} + \left(56 a + 247\right)\cdot 307^{5} + \left(238 a + 213\right)\cdot 307^{6} + \left(238 a + 73\right)\cdot 307^{7} + \left(196 a + 295\right)\cdot 307^{8} + \left(4 a + 106\right)\cdot 307^{9} +O(307^{10})$$ 303*a + 119 + (115*a + 212)*307 + (238*a + 19)*307^2 + (217*a + 39)*307^3 + (104*a + 185)*307^4 + (56*a + 247)*307^5 + (238*a + 213)*307^6 + (238*a + 73)*307^7 + (196*a + 295)*307^8 + (4*a + 106)*307^9+O(307^10) $r_{ 2 }$ $=$ $$281 a + 246 + \left(73 a + 233\right)\cdot 307 + \left(135 a + 47\right)\cdot 307^{2} + \left(43 a + 134\right)\cdot 307^{3} + \left(228 a + 141\right)\cdot 307^{4} + \left(250 a + 230\right)\cdot 307^{5} + \left(30 a + 115\right)\cdot 307^{6} + \left(262 a + 159\right)\cdot 307^{7} + \left(205 a + 62\right)\cdot 307^{8} + \left(114 a + 46\right)\cdot 307^{9} +O(307^{10})$$ 281*a + 246 + (73*a + 233)*307 + (135*a + 47)*307^2 + (43*a + 134)*307^3 + (228*a + 141)*307^4 + (250*a + 230)*307^5 + (30*a + 115)*307^6 + (262*a + 159)*307^7 + (205*a + 62)*307^8 + (114*a + 46)*307^9+O(307^10) $r_{ 3 }$ $=$ $$152 + 240\cdot 307 + 211\cdot 307^{2} + 270\cdot 307^{3} + 79\cdot 307^{4} + 60\cdot 307^{5} + 39\cdot 307^{6} + 41\cdot 307^{7} + 24\cdot 307^{8} + 10\cdot 307^{9} +O(307^{10})$$ 152 + 240*307 + 211*307^2 + 270*307^3 + 79*307^4 + 60*307^5 + 39*307^6 + 41*307^7 + 24*307^8 + 10*307^9+O(307^10) $r_{ 4 }$ $=$ $$4 a + 115 + \left(191 a + 25\right)\cdot 307 + \left(68 a + 142\right)\cdot 307^{2} + \left(89 a + 18\right)\cdot 307^{3} + \left(202 a + 72\right)\cdot 307^{4} + \left(250 a + 199\right)\cdot 307^{5} + \left(68 a + 88\right)\cdot 307^{6} + \left(68 a + 74\right)\cdot 307^{7} + \left(110 a + 253\right)\cdot 307^{8} + \left(302 a + 221\right)\cdot 307^{9} +O(307^{10})$$ 4*a + 115 + (191*a + 25)*307 + (68*a + 142)*307^2 + (89*a + 18)*307^3 + (202*a + 72)*307^4 + (250*a + 199)*307^5 + (68*a + 88)*307^6 + (68*a + 74)*307^7 + (110*a + 253)*307^8 + (302*a + 221)*307^9+O(307^10) $r_{ 5 }$ $=$ $$150 + 68\cdot 307 + 274\cdot 307^{2} + 301\cdot 307^{3} + 212\cdot 307^{4} + 16\cdot 307^{5} + 48\cdot 307^{6} + 196\cdot 307^{7} + 2\cdot 307^{8} + 223\cdot 307^{9} +O(307^{10})$$ 150 + 68*307 + 274*307^2 + 301*307^3 + 212*307^4 + 16*307^5 + 48*307^6 + 196*307^7 + 2*307^8 + 223*307^9+O(307^10) $r_{ 6 }$ $=$ $$65 + 304\cdot 307 + 293\cdot 307^{2} + 301\cdot 307^{3} + 246\cdot 307^{4} + 51\cdot 307^{5} + 187\cdot 307^{6} + 144\cdot 307^{7} + 50\cdot 307^{8} + 60\cdot 307^{9} +O(307^{10})$$ 65 + 304*307 + 293*307^2 + 301*307^3 + 246*307^4 + 51*307^5 + 187*307^6 + 144*307^7 + 50*307^8 + 60*307^9+O(307^10) $r_{ 7 }$ $=$ $$26 a + 220 + \left(233 a + 26\right)\cdot 307 + \left(171 a + 109\right)\cdot 307^{2} + \left(263 a + 42\right)\cdot 307^{3} + \left(78 a + 19\right)\cdot 307^{4} + \left(56 a + 253\right)\cdot 307^{5} + \left(276 a + 202\right)\cdot 307^{6} + \left(44 a + 83\right)\cdot 307^{7} + \left(101 a + 6\right)\cdot 307^{8} + \left(192 a + 262\right)\cdot 307^{9} +O(307^{10})$$ 26*a + 220 + (233*a + 26)*307 + (171*a + 109)*307^2 + (263*a + 42)*307^3 + (78*a + 19)*307^4 + (56*a + 253)*307^5 + (276*a + 202)*307^6 + (44*a + 83)*307^7 + (101*a + 6)*307^8 + (192*a + 262)*307^9+O(307^10) $r_{ 8 }$ $=$ $$161 + 116\cdot 307 + 129\cdot 307^{2} + 119\cdot 307^{3} + 270\cdot 307^{4} + 168\cdot 307^{5} + 25\cdot 307^{6} + 148\cdot 307^{7} + 226\cdot 307^{8} + 297\cdot 307^{9} +O(307^{10})$$ 161 + 116*307 + 129*307^2 + 119*307^3 + 270*307^4 + 168*307^5 + 25*307^6 + 148*307^7 + 226*307^8 + 297*307^9+O(307^10)

## 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.