Basic invariants
| Dimension: | $45$ |
| Group: | $A_8$ |
| Conductor: | \(703\!\cdots\!232\)\(\medspace = 2^{130} \cdot 23^{39} \cdot 35809^{39} \) |
| Artin stem field: | Galois closure of 8.0.5113757317969899544771546325450569302016.1 |
| 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 8.0.5113757317969899544771546325450569302016.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823600 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 797 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 797 }$:
\( x^{2} + 793x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 309 a + 652 + \left(256 a + 581\right)\cdot 797 + \left(443 a + 115\right)\cdot 797^{2} + \left(138 a + 488\right)\cdot 797^{3} + \left(287 a + 153\right)\cdot 797^{4} + \left(540 a + 411\right)\cdot 797^{5} + \left(125 a + 665\right)\cdot 797^{6} + \left(454 a + 27\right)\cdot 797^{7} + \left(565 a + 489\right)\cdot 797^{8} + \left(464 a + 695\right)\cdot 797^{9} +O(797^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 731 a + 319 + \left(445 a + 368\right)\cdot 797 + \left(75 a + 425\right)\cdot 797^{2} + \left(622 a + 516\right)\cdot 797^{3} + \left(673 a + 330\right)\cdot 797^{4} + \left(218 a + 477\right)\cdot 797^{5} + \left(183 a + 390\right)\cdot 797^{6} + \left(133 a + 116\right)\cdot 797^{7} + \left(131 a + 631\right)\cdot 797^{8} + \left(171 a + 767\right)\cdot 797^{9} +O(797^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 488 a + 294 + \left(540 a + 501\right)\cdot 797 + \left(353 a + 38\right)\cdot 797^{2} + \left(658 a + 599\right)\cdot 797^{3} + \left(509 a + 366\right)\cdot 797^{4} + \left(256 a + 691\right)\cdot 797^{5} + \left(671 a + 627\right)\cdot 797^{6} + \left(342 a + 124\right)\cdot 797^{7} + \left(231 a + 703\right)\cdot 797^{8} + \left(332 a + 394\right)\cdot 797^{9} +O(797^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 66 a + 55 + \left(351 a + 624\right)\cdot 797 + \left(721 a + 281\right)\cdot 797^{2} + \left(174 a + 538\right)\cdot 797^{3} + \left(123 a + 12\right)\cdot 797^{4} + \left(578 a + 679\right)\cdot 797^{5} + \left(613 a + 107\right)\cdot 797^{6} + \left(663 a + 466\right)\cdot 797^{7} + \left(665 a + 225\right)\cdot 797^{8} + \left(625 a + 524\right)\cdot 797^{9} +O(797^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 739 + 21\cdot 797 + 675\cdot 797^{2} + 210\cdot 797^{3} + 269\cdot 797^{4} + 123\cdot 797^{5} + 105\cdot 797^{6} + 226\cdot 797^{7} + 440\cdot 797^{8} + 363\cdot 797^{9} +O(797^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 265 + 682\cdot 797 + 623\cdot 797^{2} + 759\cdot 797^{3} + 247\cdot 797^{4} + 498\cdot 797^{5} + 282\cdot 797^{6} + 193\cdot 797^{7} + 66\cdot 797^{8} + 389\cdot 797^{9} +O(797^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 675 + 656\cdot 797 + 483\cdot 797^{2} + 333\cdot 797^{3} + 673\cdot 797^{4} + 516\cdot 797^{5} + 79\cdot 797^{6} + 159\cdot 797^{7} + 492\cdot 797^{8} + 281\cdot 797^{9} +O(797^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 189 + 548\cdot 797 + 543\cdot 797^{2} + 538\cdot 797^{3} + 336\cdot 797^{4} + 587\cdot 797^{5} + 131\cdot 797^{6} + 280\cdot 797^{7} + 140\cdot 797^{8} + 568\cdot 797^{9} +O(797^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $45$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-3$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $-3$ |
| $112$ | $3$ | $(1,2,3)$ | $0$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $1$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $0$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $0$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.