Basic invariants
| Dimension: | $2$ |
| Group: | $\SL(2,3)$ |
| Conductor: | \(679\)\(\medspace = 7 \cdot 97 \) |
| Artin stem field: | Galois closure of 8.0.212558803681.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $\SL(2,3)$ |
| Parity: | even |
| Determinant: | 1.679.3t1.b.b |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.4.461041.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 15x^{6} + 60x^{4} + 35x^{2} + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{3} + 2x + 11 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a^{2} + 7 a + 11 + \left(11 a^{2} + 6 a + 3\right)\cdot 13 + \left(4 a^{2} + 8 a + 1\right)\cdot 13^{2} + \left(4 a + 6\right)\cdot 13^{3} + \left(7 a^{2} + 3\right)\cdot 13^{4} + \left(6 a^{2} + 9 a + 12\right)\cdot 13^{5} + \left(a^{2} + 6 a + 3\right)\cdot 13^{6} + \left(5 a^{2} + 11 a + 11\right)\cdot 13^{7} + \left(11 a^{2} + 7 a + 6\right)\cdot 13^{8} + \left(12 a + 10\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 10 + \left(12 a^{2} + 6 a + 5\right)\cdot 13 + \left(11 a^{2} + 6 a + 12\right)\cdot 13^{2} + \left(9 a^{2} + 2 a + 11\right)\cdot 13^{3} + \left(a^{2} + 8 a + 3\right)\cdot 13^{4} + \left(12 a^{2} + 11 a + 8\right)\cdot 13^{5} + \left(9 a^{2} + 10 a + 2\right)\cdot 13^{6} + \left(a^{2} + 9 a + 2\right)\cdot 13^{7} + \left(10 a^{2} + 5 a\right)\cdot 13^{8} + \left(2 a^{2} + 8 a + 3\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 + 5\cdot 13 + 4\cdot 13^{2} + 11\cdot 13^{3} + 5\cdot 13^{4} + 4\cdot 13^{5} + 11\cdot 13^{6} + 13^{7} + 3\cdot 13^{8} + 2\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a^{2} + 2 a + 8 + 6\cdot 13 + \left(7 a^{2} + 11 a + 8\right)\cdot 13^{2} + \left(9 a^{2} + 10 a + 9\right)\cdot 13^{3} + \left(7 a^{2} + 7 a + 8\right)\cdot 13^{4} + \left(5 a^{2} + 2 a + 6\right)\cdot 13^{5} + \left(8 a^{2} + 4 a + 4\right)\cdot 13^{6} + \left(9 a^{2} + 11 a + 4\right)\cdot 13^{7} + \left(11 a^{2} + 10 a + 7\right)\cdot 13^{8} + \left(a^{2} + 8 a + 7\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a^{2} + 6 a + 2 + \left(a^{2} + 6 a + 9\right)\cdot 13 + \left(8 a^{2} + 4 a + 11\right)\cdot 13^{2} + \left(12 a^{2} + 8 a + 6\right)\cdot 13^{3} + \left(5 a^{2} + 12 a + 9\right)\cdot 13^{4} + \left(6 a^{2} + 3 a\right)\cdot 13^{5} + \left(11 a^{2} + 6 a + 9\right)\cdot 13^{6} + \left(7 a^{2} + a + 1\right)\cdot 13^{7} + \left(a^{2} + 5 a + 6\right)\cdot 13^{8} + \left(12 a^{2} + 2\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 3 + \left(a^{2} + 6 a + 7\right)\cdot 13 + \left(a^{2} + 6 a\right)\cdot 13^{2} + \left(3 a^{2} + 10 a + 1\right)\cdot 13^{3} + \left(11 a^{2} + 4 a + 9\right)\cdot 13^{4} + \left(a + 4\right)\cdot 13^{5} + \left(3 a^{2} + 2 a + 10\right)\cdot 13^{6} + \left(11 a^{2} + 3 a + 10\right)\cdot 13^{7} + \left(2 a^{2} + 7 a + 12\right)\cdot 13^{8} + \left(10 a^{2} + 4 a + 9\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 10 + 7\cdot 13 + 8\cdot 13^{2} + 13^{3} + 7\cdot 13^{4} + 8\cdot 13^{5} + 13^{6} + 11\cdot 13^{7} + 9\cdot 13^{8} + 10\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 6 a^{2} + 11 a + 5 + \left(12 a^{2} + 12 a + 6\right)\cdot 13 + \left(5 a^{2} + a + 4\right)\cdot 13^{2} + \left(3 a^{2} + 2 a + 3\right)\cdot 13^{3} + \left(5 a^{2} + 5 a + 4\right)\cdot 13^{4} + \left(7 a^{2} + 10 a + 6\right)\cdot 13^{5} + \left(4 a^{2} + 8 a + 8\right)\cdot 13^{6} + \left(3 a^{2} + a + 8\right)\cdot 13^{7} + \left(a^{2} + 2 a + 5\right)\cdot 13^{8} + \left(11 a^{2} + 4 a + 5\right)\cdot 13^{9} +O(13^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $4$ | $3$ | $(1,7,2)(3,6,5)$ | $-\zeta_{3}$ |
| $4$ | $3$ | $(1,2,7)(3,5,6)$ | $\zeta_{3} + 1$ |
| $6$ | $4$ | $(1,7,5,3)(2,4,6,8)$ | $0$ |
| $4$ | $6$ | $(1,6,7,5,2,3)(4,8)$ | $-\zeta_{3} - 1$ |
| $4$ | $6$ | $(1,3,2,5,7,6)(4,8)$ | $\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.