Basic invariants
| Dimension: | $3$ |
| Group: | $\GL(3,2)$ |
| Conductor: | \(3225800\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 127^{2} \) |
| Artin stem field: | Galois closure of 7.3.25806400.2 |
| 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 7.3.25806400.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - 3x^{5} + 7x^{4} - 7x^{3} - x^{2} + 3x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{3} + 2x + 11 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a^{2} + 6 a + 12 + \left(5 a^{2} + 10 a + 12\right)\cdot 13 + \left(12 a^{2} + 6 a + 12\right)\cdot 13^{2} + \left(11 a + 5\right)\cdot 13^{3} + \left(8 a^{2} + 6 a + 10\right)\cdot 13^{4} + \left(7 a^{2} + 11 a + 1\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a^{2} + 4 a + 7 + \left(8 a^{2} + 5\right)\cdot 13 + \left(12 a^{2} + 12 a + 11\right)\cdot 13^{2} + \left(2 a^{2} + 2 a + 10\right)\cdot 13^{3} + \left(8 a^{2} + 2 a + 2\right)\cdot 13^{4} + \left(12 a^{2} + 7 a + 3\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 8 + \left(9 a^{2} + 3 a + 1\right)\cdot 13 + \left(12 a^{2} + 9 a + 7\right)\cdot 13^{2} + \left(a + 12\right)\cdot 13^{3} + \left(11 a^{2} + 2 a + 10\right)\cdot 13^{4} + \left(8 a^{2} + 12 a + 6\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a^{2} + 3 a + 9 + \left(8 a^{2} + 9 a\right)\cdot 13 + \left(4 a + 4\right)\cdot 13^{2} + \left(9 a^{2} + 8 a + 10\right)\cdot 13^{3} + \left(6 a^{2} + 8 a\right)\cdot 13^{4} + \left(4 a^{2} + 6 a + 1\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 + 13^{2} + 4\cdot 13^{3} + 12\cdot 13^{4} +O(13^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 11 a^{2} + 12 a + 4 + \left(11 a^{2} + 8 a + 4\right)\cdot 13 + \left(5 a^{2} + a + 4\right)\cdot 13^{2} + \left(4 a^{2} + 4 a + 6\right)\cdot 13^{3} + \left(10 a^{2} + 9 a\right)\cdot 13^{4} + \left(9 a^{2} + a + 9\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 11 a^{2} + 8 a + 4 + \left(8 a^{2} + 6 a\right)\cdot 13 + \left(7 a^{2} + 4 a + 11\right)\cdot 13^{2} + \left(7 a^{2} + 10 a + 1\right)\cdot 13^{3} + \left(7 a^{2} + 9 a + 1\right)\cdot 13^{4} + \left(8 a^{2} + 12 a + 3\right)\cdot 13^{5} +O(13^{6})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $21$ | $2$ | $(1,2)(3,5)$ | $-1$ |
| $56$ | $3$ | $(2,6,5)(3,4,7)$ | $0$ |
| $42$ | $4$ | $(1,4)(2,7,6,3)$ | $1$ |
| $24$ | $7$ | $(1,7,6,3,5,2,4)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $24$ | $7$ | $(1,3,4,6,2,7,5)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.