Basic invariants
| Dimension: | $7$ |
| Group: | $C_2^3:(C_7: C_3)$ |
| Conductor: | \(1135929640000\)\(\medspace = 2^{6} \cdot 5^{4} \cdot 73^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.1135929640000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2^3:(C_7: C_3)$ |
| Parity: | even |
| Projective image: | $F_8:C_3$ |
| Projective field: | Galois closure of 8.0.1135929640000.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( a^{2} + 2 a + 10 + \left(10 a^{2} + 5 a + 10\right)\cdot 11 + \left(2 a^{2} + 2 a + 6\right)\cdot 11^{2} + \left(5 a^{2} + 6 a + 8\right)\cdot 11^{3} + \left(5 a^{2} + 4 a + 4\right)\cdot 11^{4} + \left(9 a^{2} + 4 a + 5\right)\cdot 11^{5} + \left(8 a^{2} + 4 a + 5\right)\cdot 11^{6} + \left(a^{2} + 10 a + 3\right)\cdot 11^{7} + \left(6 a^{2} + 5 a + 2\right)\cdot 11^{8} + 10 a^{2} 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 + 3\cdot 11 + 4\cdot 11^{2} + 4\cdot 11^{3} + 2\cdot 11^{4} + 5\cdot 11^{6} + 8\cdot 11^{7} + 3\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 8 + \left(6 a^{2} + 8 a\right)\cdot 11 + \left(2 a^{2} + 9 a\right)\cdot 11^{2} + \left(3 a^{2} + a + 5\right)\cdot 11^{3} + \left(6 a^{2} + 7 a + 8\right)\cdot 11^{4} + \left(10 a^{2} + 8 a + 3\right)\cdot 11^{5} + \left(7 a^{2} + 2 a + 3\right)\cdot 11^{6} + \left(5 a + 1\right)\cdot 11^{7} + \left(8 a^{2} + 9 a + 3\right)\cdot 11^{8} + \left(3 a^{2} + 4 a + 2\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a^{2} + 6 a + 5 + \left(3 a^{2} + 7 a + 8\right)\cdot 11 + \left(5 a^{2} + 3\right)\cdot 11^{2} + \left(3 a^{2} + 8 a + 5\right)\cdot 11^{3} + \left(2 a^{2} + 10\right)\cdot 11^{4} + \left(2 a^{2} + 4 a + 10\right)\cdot 11^{5} + \left(10 a^{2} + 7 a + 9\right)\cdot 11^{6} + \left(6 a^{2} + 7 a + 5\right)\cdot 11^{7} + \left(10 a^{2} + 6\right)\cdot 11^{8} + \left(9 a^{2} + 6 a + 10\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 + 4\cdot 11 + 7\cdot 11^{2} + 10\cdot 11^{3} + 4\cdot 11^{4} + 8\cdot 11^{5} + 2\cdot 11^{6} + 9\cdot 11^{7} + 6\cdot 11^{8} + 2\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a^{2} + 10 a + \left(10 a^{2} + 1\right)\cdot 11 + \left(a + 8\right)\cdot 11^{2} + \left(8 a^{2} + 7 a + 8\right)\cdot 11^{3} + \left(9 a^{2} + a + 6\right)\cdot 11^{4} + \left(6 a^{2} + 9 a + 5\right)\cdot 11^{5} + \left(2 a + 5\right)\cdot 11^{6} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{7} + \left(10 a^{2} + 4 a + 4\right)\cdot 11^{8} + \left(2 a^{2} + 9 a + 8\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 10 a + 5 + \left(a^{2} + 4 a + 2\right)\cdot 11 + \left(7 a^{2} + 7 a + 5\right)\cdot 11^{2} + \left(8 a^{2} + 8 a + 9\right)\cdot 11^{3} + \left(6 a^{2} + 4 a + 2\right)\cdot 11^{4} + \left(5 a^{2} + 8 a\right)\cdot 11^{5} + \left(a^{2} + 3 a + 3\right)\cdot 11^{6} + \left(9 a + 1\right)\cdot 11^{7} + \left(5 a^{2} + 8\right)\cdot 11^{8} + \left(8 a^{2} + a + 4\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 5 a^{2} + 10 a + \left(a^{2} + 5 a + 2\right)\cdot 11 + \left(3 a^{2} + 8\right)\cdot 11^{2} + \left(4 a^{2} + a + 2\right)\cdot 11^{3} + \left(2 a^{2} + 3 a + 3\right)\cdot 11^{4} + \left(9 a^{2} + 9 a + 9\right)\cdot 11^{5} + \left(3 a^{2} + 8\right)\cdot 11^{6} + \left(3 a^{2} + 9 a + 4\right)\cdot 11^{7} + 3 a^{2} 11^{8} + \left(8 a^{2} + 1\right)\cdot 11^{9} +O(11^{10})\)
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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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $7$ |
| $7$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $-1$ |
| $28$ | $3$ | $(1,2,3)(6,8,7)$ | $1$ |
| $28$ | $3$ | $(1,3,2)(6,7,8)$ | $1$ |
| $28$ | $6$ | $(1,8,2,7,3,6)(4,5)$ | $-1$ |
| $28$ | $6$ | $(1,6,3,7,2,8)(4,5)$ | $-1$ |
| $24$ | $7$ | $(1,7,8,6,4,3,5)$ | $0$ |
| $24$ | $7$ | $(1,6,5,8,3,7,4)$ | $0$ |