Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(680\)\(\medspace = 2^{3} \cdot 5 \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 3.1.680.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Determinant: | 1.680.2t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.680.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - x^{2} - 6x + 10 \)
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The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 24 + 10\cdot 79 + 18\cdot 79^{2} + 20\cdot 79^{3} + 33\cdot 79^{4} +O(79^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 27 + 66\cdot 79 + 22\cdot 79^{2} + 47\cdot 79^{3} + 39\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 + 2\cdot 79 + 38\cdot 79^{2} + 11\cdot 79^{3} + 6\cdot 79^{4} +O(79^{5})\)
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Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,2)$ | $0$ | ✓ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |