Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(43659\)\(\medspace = 3^{4} \cdot 7^{2} \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.173282571.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.11.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.480249.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 15x^{4} - 7x^{3} + 54x^{2} + 21x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 3\cdot 23 + 12\cdot 23^{2} + 14\cdot 23^{3} + 23^{4} + 22\cdot 23^{5} + 18\cdot 23^{6} + 15\cdot 23^{7} + 6\cdot 23^{8} +O(23^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a + 19 + \left(15 a + 8\right)\cdot 23 + \left(3 a + 21\right)\cdot 23^{2} + \left(7 a + 14\right)\cdot 23^{3} + \left(11 a + 6\right)\cdot 23^{4} + 13\cdot 23^{5} + \left(18 a + 1\right)\cdot 23^{6} + \left(19 a + 17\right)\cdot 23^{7} + \left(2 a + 11\right)\cdot 23^{8} +O(23^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 13 + \left(7 a + 20\right)\cdot 23 + \left(19 a + 12\right)\cdot 23^{2} + \left(15 a + 2\right)\cdot 23^{3} + \left(11 a + 22\right)\cdot 23^{4} + \left(22 a + 2\right)\cdot 23^{5} + \left(4 a + 14\right)\cdot 23^{6} + \left(3 a + 15\right)\cdot 23^{7} + \left(20 a + 20\right)\cdot 23^{8} +O(23^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 5\cdot 23 + 19\cdot 23^{2} + 8\cdot 23^{3} + 7\cdot 23^{4} + 15\cdot 23^{5} + 8\cdot 23^{6} + 9\cdot 23^{7} + 23^{8} +O(23^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 14 + \left(15 a + 15\right)\cdot 23 + \left(7 a + 1\right)\cdot 23^{2} + \left(10 a + 19\right)\cdot 23^{3} + \left(21 a + 10\right)\cdot 23^{4} + \left(6 a + 11\right)\cdot 23^{5} + \left(2 a + 2\right)\cdot 23^{6} + \left(12 a + 6\right)\cdot 23^{7} + \left(7 a + 1\right)\cdot 23^{8} +O(23^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a + 7 + \left(7 a + 15\right)\cdot 23 + \left(15 a + 1\right)\cdot 23^{2} + \left(12 a + 9\right)\cdot 23^{3} + \left(a + 20\right)\cdot 23^{4} + \left(16 a + 3\right)\cdot 23^{5} + 20 a\cdot 23^{6} + \left(10 a + 5\right)\cdot 23^{7} + \left(15 a + 4\right)\cdot 23^{8} +O(23^{9})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ | |
| $3$ | $2$ | $(1,4)$ | $1$ | ✓ |
| $3$ | $2$ | $(1,4)(5,6)$ | $-1$ | |
| $4$ | $3$ | $(1,2,5)(3,6,4)$ | $0$ | |
| $4$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ | |
| $4$ | $6$ | $(1,3,6,4,2,5)$ | $0$ | |
| $4$ | $6$ | $(1,5,2,4,6,3)$ | $0$ |