Basic invariants
| Dimension: | $4$ |
| Group: | $S_3^2$ |
| Conductor: | \(54523456\)\(\medspace = 2^{6} \cdot 13^{2} \cdot 71^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.50325149888.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3^2$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3^2$ |
| Projective stem field: | Galois closure of 6.0.50325149888.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 13x^{4} - 4x^{3} + 273x^{2} + 26x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{2} + 7x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a + 1 + \left(6 a + 8\right)\cdot 11 + \left(8 a + 10\right)\cdot 11^{2} + \left(9 a + 1\right)\cdot 11^{3} + \left(8 a + 8\right)\cdot 11^{4} + \left(7 a + 8\right)\cdot 11^{5} + \left(5 a + 10\right)\cdot 11^{6} + \left(8 a + 3\right)\cdot 11^{7} + \left(6 a + 9\right)\cdot 11^{8} + \left(4 a + 7\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 6\cdot 11 + 6\cdot 11^{2} + 11^{3} + 6\cdot 11^{4} + 5\cdot 11^{5} + 9\cdot 11^{6} + 5\cdot 11^{7} + 3\cdot 11^{8} + 6\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 + 4\cdot 11 + 5\cdot 11^{2} + 9\cdot 11^{3} + 11^{4} + 4\cdot 11^{5} + 7\cdot 11^{6} + 7\cdot 11^{7} + 6\cdot 11^{8} + 5\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 5 + \left(4 a + 9\right)\cdot 11 + \left(2 a + 5\right)\cdot 11^{2} + \left(a + 10\right)\cdot 11^{3} + 2 a\cdot 11^{4} + \left(3 a + 9\right)\cdot 11^{5} + \left(5 a + 3\right)\cdot 11^{6} + \left(2 a + 10\right)\cdot 11^{7} + \left(4 a + 5\right)\cdot 11^{8} + \left(6 a + 8\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 5 + \left(6 a + 1\right)\cdot 11 + \left(5 a + 5\right)\cdot 11^{2} + \left(4 a + 9\right)\cdot 11^{3} + \left(9 a + 7\right)\cdot 11^{4} + \left(9 a + 9\right)\cdot 11^{5} + \left(5 a + 4\right)\cdot 11^{6} + \left(5 a + 5\right)\cdot 11^{7} + 5\cdot 11^{8} + \left(3 a + 7\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 9 + \left(4 a + 2\right)\cdot 11 + \left(5 a + 10\right)\cdot 11^{2} + \left(6 a + 10\right)\cdot 11^{3} + \left(a + 7\right)\cdot 11^{4} + \left(a + 6\right)\cdot 11^{5} + \left(5 a + 7\right)\cdot 11^{6} + \left(5 a + 10\right)\cdot 11^{7} + \left(10 a + 1\right)\cdot 11^{8} + \left(7 a + 8\right)\cdot 11^{9} +O(11^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ | |
| $3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ | ✓ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ | |
| $9$ | $2$ | $(2,6)(3,4)$ | $0$ | |
| $2$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ | |
| $2$ | $3$ | $(1,3,4)(2,6,5)$ | $-2$ | |
| $4$ | $3$ | $(1,4,3)$ | $1$ | |
| $6$ | $6$ | $(1,2,3,5,4,6)$ | $0$ | |
| $6$ | $6$ | $(1,5,3,2,4,6)$ | $0$ |