Basic invariants
| Dimension: | $28$ |
| Group: | $A_8$ |
| Conductor: | \(136\!\cdots\!736\)\(\medspace = 2^{90} \cdot 7^{34} \cdot 268913^{24} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.36574214064047828349270556528863627894423814144.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 56 |
| Parity: | even |
| Projective image: | $A_8$ |
| Projective field: | Galois closure of 8.0.36574214064047828349270556528863627894423814144.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$:
\( x^{3} + 3x + 81 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 29 a^{2} + 75 a + 67 + \left(5 a^{2} + 15 a + 15\right)\cdot 83 + \left(19 a^{2} + 56 a + 63\right)\cdot 83^{2} + \left(32 a^{2} + 77 a + 44\right)\cdot 83^{3} + \left(57 a^{2} + 48 a + 71\right)\cdot 83^{4} + \left(64 a^{2} + 78 a + 23\right)\cdot 83^{5} + \left(4 a^{2} + 79 a + 12\right)\cdot 83^{6} + \left(74 a^{2} + 2 a + 50\right)\cdot 83^{7} + \left(23 a^{2} + 46 a + 26\right)\cdot 83^{8} + \left(72 a^{2} + 30 a + 49\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 38 a^{2} + 81 a + 57 + \left(34 a^{2} + 16 a + 40\right)\cdot 83 + \left(33 a^{2} + 14 a + 12\right)\cdot 83^{2} + \left(12 a^{2} + 37 a + 10\right)\cdot 83^{3} + \left(42 a^{2} + 78 a + 49\right)\cdot 83^{4} + \left(34 a^{2} + 21 a + 82\right)\cdot 83^{5} + \left(55 a^{2} + 74 a + 2\right)\cdot 83^{6} + \left(66 a^{2} + 28 a + 52\right)\cdot 83^{7} + \left(56 a^{2} + 27 a + 17\right)\cdot 83^{8} + \left(80 a^{2} + a + 66\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 72 a^{2} + 27 a + 42 + \left(8 a^{2} + 25 a + 72\right)\cdot 83 + \left(56 a^{2} + 29 a + 57\right)\cdot 83^{2} + \left(63 a^{2} + 8 a + 29\right)\cdot 83^{3} + \left(35 a + 49\right)\cdot 83^{4} + \left(76 a^{2} + 21 a + 82\right)\cdot 83^{5} + \left(70 a^{2} + 44 a + 33\right)\cdot 83^{6} + \left(16 a^{2} + 26 a + 35\right)\cdot 83^{7} + \left(40 a^{2} + 22 a + 67\right)\cdot 83^{8} + \left(49 a^{2} + 80 a + 3\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 56 a^{2} + 58 a + 10 + \left(39 a^{2} + 40 a + 51\right)\cdot 83 + \left(76 a^{2} + 39 a + 15\right)\cdot 83^{2} + \left(6 a^{2} + 37 a + 82\right)\cdot 83^{3} + \left(40 a^{2} + 52 a + 44\right)\cdot 83^{4} + \left(55 a^{2} + 39 a + 41\right)\cdot 83^{5} + \left(39 a^{2} + 47 a + 54\right)\cdot 83^{6} + \left(82 a^{2} + 27 a\right)\cdot 83^{7} + \left(68 a^{2} + 33 a + 42\right)\cdot 83^{8} + \left(35 a^{2} + a + 59\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 71 + 71\cdot 83 + 63\cdot 83^{2} + 40\cdot 83^{3} + 12\cdot 83^{4} + 52\cdot 83^{5} + 77\cdot 83^{6} + 53\cdot 83^{7} + 14\cdot 83^{8} + 19\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a^{2} + 58 a + 15 + \left(31 a^{2} + 75 a + 67\right)\cdot 83 + \left(a^{2} + 29 a + 27\right)\cdot 83^{2} + \left(10 a^{2} + 29 a\right)\cdot 83^{3} + \left(34 a^{2} + 74 a + 25\right)\cdot 83^{4} + \left(39 a^{2} + 37 a + 56\right)\cdot 83^{5} + \left(53 a^{2} + a + 26\right)\cdot 83^{6} + \left(40 a^{2} + 42 a + 66\right)\cdot 83^{7} + \left(11 a^{2} + 75 a + 1\right)\cdot 83^{8} + \left(19 a^{2} + 33 a + 26\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 42 + 80\cdot 83 + 23\cdot 83^{2} + 62\cdot 83^{3} + 56\cdot 83^{4} + 57\cdot 83^{5} + 71\cdot 83^{6} + 68\cdot 83^{7} + 4\cdot 83^{8} + 54\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 51 a^{2} + 33 a + 28 + \left(46 a^{2} + 74 a + 15\right)\cdot 83 + \left(62 a^{2} + 79 a + 67\right)\cdot 83^{2} + \left(40 a^{2} + 58 a + 61\right)\cdot 83^{3} + \left(74 a^{2} + 42 a + 22\right)\cdot 83^{4} + \left(61 a^{2} + 49 a + 18\right)\cdot 83^{5} + \left(24 a^{2} + a + 52\right)\cdot 83^{6} + \left(51 a^{2} + 38 a + 4\right)\cdot 83^{7} + \left(47 a^{2} + 44 a + 74\right)\cdot 83^{8} + \left(74 a^{2} + 18 a + 53\right)\cdot 83^{9} +O(83^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $28$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-4$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $4$ |
| $112$ | $3$ | $(1,2,3)$ | $1$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $-2$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $-1$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $1$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $1$ |