Basic invariants
| Dimension: | $3$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(19456921\)\(\medspace = 11^{2} \cdot 401^{2} \) |
| Artin stem field: | Galois closure of 7.3.19456921.2 |
| 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 7.3.19456921.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 5x^{5} - 4x^{4} + 3x^{3} + 3x^{2} + 2x + 4 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 26 a^{2} + 30 a + 18 + \left(28 a^{2} + 22 a + 11\right)\cdot 31 + \left(28 a^{2} + 2 a + 10\right)\cdot 31^{2} + \left(9 a^{2} + a + 25\right)\cdot 31^{3} + \left(3 a^{2} + 21 a + 16\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + 14 + \left(28 a^{2} + 28 a + 21\right)\cdot 31 + \left(28 a^{2} + 29 a + 6\right)\cdot 31^{2} + \left(23 a^{2} + 17 a + 14\right)\cdot 31^{3} + \left(25 a^{2} + 13 a + 1\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 11 + \left(25 a^{2} + 18 a + 19\right)\cdot 31 + \left(18 a^{2} + 18 a + 3\right)\cdot 31^{2} + \left(13 a^{2} + 9 a + 7\right)\cdot 31^{3} + \left(26 a^{2} + 22 a + 1\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 a^{2} + 7 a + 15 + \left(23 a^{2} + 12 a + 18\right)\cdot 31 + \left(30 a^{2} + 27 a + 28\right)\cdot 31^{2} + \left(12 a^{2} + 25 a + 6\right)\cdot 31^{3} + \left(27 a^{2} + 3 a + 23\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a^{2} + 13 a + 4 + \left(8 a^{2} + 20 a + 8\right)\cdot 31 + \left(14 a^{2} + 9 a + 21\right)\cdot 31^{2} + \left(7 a^{2} + 20 a + 23\right)\cdot 31^{3} + \left(a^{2} + 18 a + 25\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a^{2} + 9 a + 13 + \left(10 a^{2} + 21 a + 30\right)\cdot 31 + \left(2 a^{2} + 4 a + 19\right)\cdot 31^{2} + \left(25 a^{2} + 18 a + 4\right)\cdot 31^{3} + \left(8 a^{2} + 13 a + 21\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 18 + 14\cdot 31 + 2\cdot 31^{2} + 11\cdot 31^{3} + 3\cdot 31^{4} +O(31^{5})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $21$ | $2$ | $(1,7)(2,3)$ | $-1$ |
| $56$ | $3$ | $(1,4,7)(2,5,3)$ | $0$ |
| $42$ | $4$ | $(1,3,7,2)(5,6)$ | $1$ |
| $24$ | $7$ | $(1,4,3,7,2,6,5)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $24$ | $7$ | $(1,7,5,3,6,4,2)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.