Basic invariants
Dimension: | $2$ |
Group: | $S_3$ |
Conductor: | \(1129\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 3.3.1129.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_3$ |
Parity: | even |
Projective image: | $S_3$ |
Projective field: | Galois closure of 3.3.1129.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 7 + 27\cdot 97 + 4\cdot 97^{2} + 33\cdot 97^{3} + 71\cdot 97^{4} +O(97^{5})\)
$r_{ 2 }$ |
$=$ |
\( 20 + 34\cdot 97 + 69\cdot 97^{2} + 28\cdot 97^{3} + 22\cdot 97^{4} +O(97^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 70 + 35\cdot 97 + 23\cdot 97^{2} + 35\cdot 97^{3} + 3\cdot 97^{4} +O(97^{5})\)
| |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,2)$ | $0$ |
$2$ | $3$ | $(1,2,3)$ | $-1$ |