Basic invariants
| Dimension: | $4$ |
| Group: | $((C_2 \times D_4): C_2):C_3$ |
| Conductor: | \(87880000\)\(\medspace = 2^{6} \cdot 5^{4} \cdot 13^{3} \) |
| Artin stem field: | Galois closure of 8.0.28561000000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T97 |
| Parity: | even |
| Determinant: | 1.13.3t1.a.a |
| Projective image: | $C_2^2:A_4$ |
| Projective stem field: | Galois closure of 12.4.2088270645760000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 9x^{6} - 2x^{5} + 9x^{4} - 9x^{3} - 24x^{2} + 7x + 11 \)
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The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3\cdot 11 + 7\cdot 11^{2} + 8\cdot 11^{3} + 9\cdot 11^{5} + 3\cdot 11^{6} + 11^{7} + 2\cdot 11^{8} + 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a^{2} + 8 a + 9 + \left(8 a^{2} + 6 a + 2\right)\cdot 11 + \left(2 a^{2} + 9\right)\cdot 11^{2} + \left(4 a + 1\right)\cdot 11^{3} + \left(10 a^{2} + 4 a + 6\right)\cdot 11^{4} + \left(6 a^{2} + a + 10\right)\cdot 11^{5} + \left(a^{2} + 8 a + 7\right)\cdot 11^{6} + \left(8 a^{2} + 9 a + 3\right)\cdot 11^{7} + \left(10 a^{2} + 10 a + 9\right)\cdot 11^{8} + \left(9 a^{2} + 7 a\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a^{2} + 2 a + 3 + \left(7 a^{2} + 8 a + 7\right)\cdot 11 + \left(a^{2} + 10\right)\cdot 11^{2} + \left(6 a^{2} + a + 9\right)\cdot 11^{3} + \left(2 a^{2} + 6\right)\cdot 11^{4} + \left(5 a^{2} + 4 a + 9\right)\cdot 11^{5} + \left(7 a^{2} + 5 a + 9\right)\cdot 11^{6} + \left(4 a^{2} + 8 a + 10\right)\cdot 11^{7} + \left(5 a + 4\right)\cdot 11^{8} + \left(2 a^{2} + 3 a + 10\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 + 9\cdot 11 + 5\cdot 11^{2} + 3\cdot 11^{3} + 10\cdot 11^{4} + 11^{6} + 6\cdot 11^{7} + 10\cdot 11^{8} + 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a^{2} + 5 a + 2 + \left(6 a^{2} + 6 a + 4\right)\cdot 11 + \left(2 a^{2} + 2 a + 5\right)\cdot 11^{2} + \left(5 a^{2} + 4 a + 8\right)\cdot 11^{3} + \left(9 a^{2} + 7 a + 1\right)\cdot 11^{4} + \left(2 a^{2} + 5\right)\cdot 11^{5} + \left(9 a^{2} + 10 a + 3\right)\cdot 11^{6} + \left(7 a + 1\right)\cdot 11^{7} + \left(2 a^{2} + 3 a + 5\right)\cdot 11^{8} + \left(9 a^{2} + 6 a + 3\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a^{2} + 3 a + 7 + \left(2 a^{2} + 7 a + 4\right)\cdot 11 + \left(10 a^{2} + 8 a + 7\right)\cdot 11^{2} + \left(2 a^{2} + 10 a + 5\right)\cdot 11^{3} + \left(7 a^{2} + 3 a + 9\right)\cdot 11^{4} + \left(10 a^{2} + 5\right)\cdot 11^{5} + \left(a + 8\right)\cdot 11^{6} + \left(4 a^{2} + 3 a + 2\right)\cdot 11^{7} + \left(8 a^{2} + 8 a + 8\right)\cdot 11^{8} + \left(4 a^{2} + 4 a + 6\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 5 a^{2} + 9 a + 10 + \left(6 a^{2} + 8 a + 10\right)\cdot 11 + \left(5 a^{2} + 7 a + 1\right)\cdot 11^{2} + \left(5 a^{2} + 2 a + 5\right)\cdot 11^{3} + \left(2 a^{2} + 10 a + 3\right)\cdot 11^{4} + \left(a^{2} + 8 a + 10\right)\cdot 11^{5} + \left(3 a + 5\right)\cdot 11^{6} + \left(2 a^{2} + 4 a + 6\right)\cdot 11^{7} + \left(9 a^{2} + 7 a + 3\right)\cdot 11^{8} + \left(2 a^{2} + 7 a + 2\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 5 a^{2} + 6 a + 4 + \left(6 a + 1\right)\cdot 11 + \left(10 a^{2} + a + 7\right)\cdot 11^{2} + \left(a^{2} + 10 a\right)\cdot 11^{3} + \left(a^{2} + 6 a + 5\right)\cdot 11^{4} + \left(6 a^{2} + 6 a + 3\right)\cdot 11^{5} + \left(2 a^{2} + 4 a + 3\right)\cdot 11^{6} + \left(2 a^{2} + 10 a\right)\cdot 11^{7} + \left(2 a^{2} + 7 a\right)\cdot 11^{8} + \left(4 a^{2} + 2 a + 6\right)\cdot 11^{9} +O(11^{10})\)
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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 value | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $1$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $-4$ | |
| $6$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $0$ | ✓ |
| $6$ | $2$ | $(1,4)(2,6)$ | $0$ | |
| $6$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $0$ | |
| $16$ | $3$ | $(1,2,7)(4,6,8)$ | $-\zeta_{3} - 1$ | |
| $16$ | $3$ | $(1,7,2)(4,8,6)$ | $\zeta_{3}$ | |
| $6$ | $4$ | $(1,2,4,6)(3,7,5,8)$ | $0$ | |
| $6$ | $4$ | $(1,5,4,3)(2,8,6,7)$ | $0$ | |
| $16$ | $6$ | $(1,8,2,4,7,6)(3,5)$ | $-\zeta_{3}$ | |
| $16$ | $6$ | $(1,6,7,4,2,8)(3,5)$ | $\zeta_{3} + 1$ |