Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(142845125\)\(\medspace = 5^{3} \cdot 1069^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.133625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | even |
| Projective image: | $\SOPlus(4,2)$ |
| Projective field: | Galois closure of 6.2.133625.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + 4 + \left(14 a + 18\right)\cdot 29 + \left(18 a + 19\right)\cdot 29^{2} + \left(3 a + 28\right)\cdot 29^{3} + \left(4 a + 9\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a + 27 + \left(5 a + 14\right)\cdot 29 + \left(12 a + 14\right)\cdot 29^{2} + \left(8 a + 21\right)\cdot 29^{3} + \left(12 a + 21\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a + 15 + \left(14 a + 23\right)\cdot 29 + \left(10 a + 10\right)\cdot 29^{2} + \left(25 a + 28\right)\cdot 29^{3} + \left(24 a + 26\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 a + 18 + \left(23 a + 7\right)\cdot 29 + \left(16 a + 12\right)\cdot 29^{2} + \left(20 a + 22\right)\cdot 29^{3} + \left(16 a + 16\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 + 6\cdot 29 + 2\cdot 29^{2} + 14\cdot 29^{3} + 19\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 + 16\cdot 29 + 27\cdot 29^{2} + 21\cdot 29^{4} +O(29^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(3,6)$ | $0$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $1$ |
| $4$ | $3$ | $(1,3,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $1$ |
| $12$ | $6$ | $(2,4,5)(3,6)$ | $0$ |