Basic invariants
| Dimension: | $45$ |
| Group: | $A_8$ |
| Conductor: | \(480\!\cdots\!168\)\(\medspace = 2^{126} \cdot 51473^{39} \) |
| Artin stem field: | Galois closure of 8.0.19501894337558159417628591379185664.1 |
| 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 8.0.19501894337558159417628591379185664.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823561 \)
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The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 59 + 21\cdot 73 + 39\cdot 73^{2} + 45\cdot 73^{3} + 46\cdot 73^{4} + 66\cdot 73^{5} + 70\cdot 73^{6} + 68\cdot 73^{7} + 59\cdot 73^{8} + 64\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 48\cdot 73 + 66\cdot 73^{2} + 60\cdot 73^{3} + 34\cdot 73^{4} + 58\cdot 73^{5} + 71\cdot 73^{6} + 71\cdot 73^{7} + 31\cdot 73^{8} + 55\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 33 + \left(69 a + 45\right)\cdot 73 + \left(47 a + 27\right)\cdot 73^{2} + \left(27 a + 47\right)\cdot 73^{3} + \left(36 a + 21\right)\cdot 73^{4} + \left(8 a + 47\right)\cdot 73^{5} + \left(15 a + 30\right)\cdot 73^{6} + \left(a + 59\right)\cdot 73^{7} + \left(68 a + 64\right)\cdot 73^{8} + \left(42 a + 13\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 69 a + 45 + \left(3 a + 29\right)\cdot 73 + \left(25 a + 29\right)\cdot 73^{2} + \left(45 a + 9\right)\cdot 73^{3} + \left(36 a + 30\right)\cdot 73^{4} + \left(64 a + 36\right)\cdot 73^{5} + \left(57 a + 67\right)\cdot 73^{6} + \left(71 a + 47\right)\cdot 73^{7} + \left(4 a + 48\right)\cdot 73^{8} + \left(30 a + 1\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 51 + 58\cdot 73 + 48\cdot 73^{2} + 11\cdot 73^{4} + 10\cdot 73^{5} + 70\cdot 73^{6} + 71\cdot 73^{7} + 22\cdot 73^{8} + 15\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 53 + 30\cdot 73 + 61\cdot 73^{2} + 60\cdot 73^{3} + 66\cdot 73^{4} + 73^{5} + 3\cdot 73^{6} + 38\cdot 73^{7} + 54\cdot 73^{8} +O(73^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 49 a + 60 + \left(38 a + 31\right)\cdot 73 + \left(12 a + 46\right)\cdot 73^{2} + 26 a\cdot 73^{3} + \left(7 a + 6\right)\cdot 73^{4} + \left(18 a + 12\right)\cdot 73^{5} + \left(50 a + 32\right)\cdot 73^{6} + \left(15 a + 41\right)\cdot 73^{7} + \left(56 a + 37\right)\cdot 73^{8} + \left(51 a + 20\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 24 a + 61 + \left(34 a + 25\right)\cdot 73 + \left(60 a + 45\right)\cdot 73^{2} + \left(46 a + 66\right)\cdot 73^{3} + \left(65 a + 1\right)\cdot 73^{4} + \left(54 a + 59\right)\cdot 73^{5} + \left(22 a + 18\right)\cdot 73^{6} + \left(57 a + 38\right)\cdot 73^{7} + \left(16 a + 44\right)\cdot 73^{8} + \left(21 a + 46\right)\cdot 73^{9} +O(73^{10})\)
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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.