Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(12064136250375\)\(\medspace = 3^{3} \cdot 5^{3} \cdot 11^{3} \cdot 139^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.22935.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PGL(2,5)$ |
Parity: | odd |
Determinant: | 1.22935.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.22935.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} + x^{3} - 2x^{2} - 4x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 7 a + 4 + \left(11 a + 10\right)\cdot 13 + \left(2 a + 5\right)\cdot 13^{2} + \left(2 a + 5\right)\cdot 13^{3} + \left(a + 9\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 2 }$ | $=$ | \( 6 a + 11 + \left(a + 1\right)\cdot 13 + \left(10 a + 10\right)\cdot 13^{2} + \left(10 a + 4\right)\cdot 13^{3} + \left(11 a + 8\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 3 }$ | $=$ | \( 3 a + 5 + \left(12 a + 3\right)\cdot 13 + 10 a\cdot 13^{2} + \left(3 a + 8\right)\cdot 13^{3} + \left(10 a + 4\right)\cdot 13^{4} +O(13^{5})\) |
$r_{ 4 }$ | $=$ | \( 11 + 10\cdot 13 + 10\cdot 13^{2} + 6\cdot 13^{3} + 5\cdot 13^{4} +O(13^{5})\) |
$r_{ 5 }$ | $=$ | \( 10 a + 8 + 12\cdot 13 + \left(2 a + 11\right)\cdot 13^{2} + 9 a\cdot 13^{3} + \left(2 a + 11\right)\cdot 13^{4} +O(13^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value |
$1$ | $1$ | $()$ | $5$ |
$10$ | $2$ | $(1,2)$ | $-1$ |
$15$ | $2$ | $(1,2)(3,4)$ | $1$ |
$20$ | $3$ | $(1,2,3)$ | $-1$ |
$30$ | $4$ | $(1,2,3,4)$ | $1$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.