# Properties

 Label 56.514...576.105.a.a Dimension $56$ Group $A_8$ Conductor $5.146\times 10^{388}$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $56$ Group: $A_8$ Conductor: $$514\!\cdots\!576$$$$\medspace = 2^{156} \cdot 23^{48} \cdot 572903^{48}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 8.0.87815967535129608031908549089667529769029306679296.1 Galois orbit size: $1$ Smallest permutation container: 105 Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_8$ Projective stem field: 8.0.87815967535129608031908549089667529769029306679296.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 112 x^{6} - 896 x^{5} - 3360 x^{4} - 7168 x^{3} - 8960 x^{2} - 6144 x + 210826512$$  .

The roots of $f$ are computed in an extension of $\Q_{ 263 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 263 }$: $$x^{2} + 261 x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$210 + 104\cdot 263 + 86\cdot 263^{2} + 48\cdot 263^{3} + 214\cdot 263^{4} + 111\cdot 263^{5} + 131\cdot 263^{6} + 248\cdot 263^{7} + 193\cdot 263^{8} + 16\cdot 263^{9} +O(263^{10})$$ $r_{ 2 }$ $=$ $$75 a + 168 + \left(192 a + 132\right)\cdot 263 + \left(217 a + 99\right)\cdot 263^{2} + \left(35 a + 65\right)\cdot 263^{3} + \left(141 a + 160\right)\cdot 263^{4} + \left(211 a + 25\right)\cdot 263^{5} + \left(126 a + 103\right)\cdot 263^{6} + \left(170 a + 99\right)\cdot 263^{7} + \left(142 a + 158\right)\cdot 263^{8} + \left(249 a + 136\right)\cdot 263^{9} +O(263^{10})$$ $r_{ 3 }$ $=$ $$45 + 43\cdot 263 + 214\cdot 263^{2} + 253\cdot 263^{3} + 80\cdot 263^{4} + 21\cdot 263^{5} + 169\cdot 263^{6} + 147\cdot 263^{7} + 211\cdot 263^{8} + 189\cdot 263^{9} +O(263^{10})$$ $r_{ 4 }$ $=$ $$179 a + 69 + \left(43 a + 73\right)\cdot 263 + \left(260 a + 36\right)\cdot 263^{2} + \left(232 a + 184\right)\cdot 263^{3} + \left(131 a + 158\right)\cdot 263^{4} + \left(166 a + 99\right)\cdot 263^{5} + \left(134 a + 57\right)\cdot 263^{6} + \left(255 a + 236\right)\cdot 263^{7} + \left(107 a + 146\right)\cdot 263^{8} + \left(68 a + 17\right)\cdot 263^{9} +O(263^{10})$$ $r_{ 5 }$ $=$ $$84 a + 164 + \left(219 a + 244\right)\cdot 263 + \left(2 a + 249\right)\cdot 263^{2} + \left(30 a + 126\right)\cdot 263^{3} + \left(131 a + 189\right)\cdot 263^{4} + \left(96 a + 37\right)\cdot 263^{5} + \left(128 a + 160\right)\cdot 263^{6} + \left(7 a + 86\right)\cdot 263^{7} + \left(155 a + 107\right)\cdot 263^{8} + \left(194 a + 46\right)\cdot 263^{9} +O(263^{10})$$ $r_{ 6 }$ $=$ $$184 + 217\cdot 263 + 6\cdot 263^{2} + 92\cdot 263^{3} + 48\cdot 263^{4} + 92\cdot 263^{5} + 262\cdot 263^{6} + 65\cdot 263^{7} + 47\cdot 263^{8} + 2\cdot 263^{9} +O(263^{10})$$ $r_{ 7 }$ $=$ $$188 a + 55 + \left(70 a + 179\right)\cdot 263 + \left(45 a + 79\right)\cdot 263^{2} + \left(227 a + 182\right)\cdot 263^{3} + \left(121 a + 143\right)\cdot 263^{4} + \left(51 a + 44\right)\cdot 263^{5} + \left(136 a + 145\right)\cdot 263^{6} + \left(92 a + 50\right)\cdot 263^{7} + \left(120 a + 10\right)\cdot 263^{8} + \left(13 a + 230\right)\cdot 263^{9} +O(263^{10})$$ $r_{ 8 }$ $=$ $$157 + 56\cdot 263 + 16\cdot 263^{2} + 99\cdot 263^{3} + 56\cdot 263^{4} + 93\cdot 263^{5} + 23\cdot 263^{6} + 117\cdot 263^{7} + 176\cdot 263^{8} + 149\cdot 263^{9} +O(263^{10})$$

## 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 $1$ $1$ $()$ $56$ $105$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $8$ $210$ $2$ $(1,2)(3,4)$ $0$ $112$ $3$ $(1,2,3)$ $-4$ $1120$ $3$ $(1,2,3)(4,5,6)$ $-1$ $1260$ $4$ $(1,2,3,4)(5,6,7,8)$ $0$ $2520$ $4$ $(1,2,3,4)(5,6)$ $0$ $1344$ $5$ $(1,2,3,4,5)$ $1$ $1680$ $6$ $(1,2,3)(4,5)(6,7)$ $0$ $3360$ $6$ $(1,2,3,4,5,6)(7,8)$ $-1$ $2880$ $7$ $(1,2,3,4,5,6,7)$ $0$ $2880$ $7$ $(1,3,4,5,6,7,2)$ $0$ $1344$ $15$ $(1,2,3,4,5)(6,7,8)$ $1$ $1344$ $15$ $(1,3,4,5,2)(6,7,8)$ $1$

The blue line marks the conjugacy class containing complex conjugation.