Basic invariants
Dimension: | $3$ |
Group: | $\GL(3,2)$ |
Conductor: | \(19456921\)\(\medspace = 11^{2} \cdot 401^{2} \) |
Artin stem field: | Galois closure of 7.3.19456921.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $\PSL(2,7)$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $\GL(3,2)$ |
Projective stem field: | Galois closure of 7.3.19456921.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 3x^{6} + 5x^{4} - 2x^{3} - 2x^{2} + 4x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{3} + x + 28 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 12 a^{2} + 26 a + 7 + \left(2 a^{2} + 2 a + 15\right)\cdot 31 + \left(3 a^{2} + 20 a + 5\right)\cdot 31^{2} + \left(10 a^{2} + 24 a + 6\right)\cdot 31^{3} + \left(22 a^{2} + 28 a + 3\right)\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$ |
$=$ |
\( 15 a^{2} + 10 a + 4 + \left(15 a^{2} + 24 a + 1\right)\cdot 31 + \left(11 a^{2} + 12 a + 11\right)\cdot 31^{2} + \left(7 a^{2} + 25 a + 10\right)\cdot 31^{3} + \left(9 a^{2} + 5 a + 1\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 24 + 17\cdot 31 + 10\cdot 31^{2} + 16\cdot 31^{3} + 18\cdot 31^{4} +O(31^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 16 a^{2} + 27 a + 20 + \left(10 a^{2} + a + 20\right)\cdot 31 + \left(10 a^{2} + 20 a + 20\right)\cdot 31^{2} + \left(15 a^{2} + a + 9\right)\cdot 31^{3} + \left(5 a^{2} + 3 a + 2\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 10 a^{2} + 11 + \left(15 a^{2} + 17 a + 11\right)\cdot 31 + \left(5 a^{2} + 8 a + 17\right)\cdot 31^{2} + \left(24 a^{2} + 17 a + 21\right)\cdot 31^{3} + \left(26 a^{2} + 10 a + 2\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 6 a^{2} + 21 a + 29 + \left(20 a + 21\right)\cdot 31 + \left(14 a^{2} + 9 a + 12\right)\cdot 31^{2} + \left(30 a^{2} + 19 a + 15\right)\cdot 31^{3} + \left(25 a^{2} + 14 a + 12\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 3 a^{2} + 9 a + 1 + \left(18 a^{2} + 26 a + 5\right)\cdot 31 + \left(17 a^{2} + 21 a + 15\right)\cdot 31^{2} + \left(5 a^{2} + 4 a + 13\right)\cdot 31^{3} + \left(3 a^{2} + 30 a + 21\right)\cdot 31^{4} +O(31^{5})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character value |
$1$ | $1$ | $()$ | $3$ |
$21$ | $2$ | $(1,4)(5,6)$ | $-1$ |
$56$ | $3$ | $(1,7,6)(2,5,3)$ | $0$ |
$42$ | $4$ | $(1,2,7,5)(3,4)$ | $1$ |
$24$ | $7$ | $(1,2,7,6,5,4,3)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
$24$ | $7$ | $(1,6,3,7,4,2,5)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.