Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(526014225\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 11^{2} \cdot 139^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 5.3.22935.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Projective image: | $S_5$ |
Projective field: | Galois closure of 5.3.22935.1 |
Galois action
Roots of defining polynomial
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $5$ |
$10$ | $2$ | $(1,2)$ | $1$ |
$15$ | $2$ | $(1,2)(3,4)$ | $1$ |
$20$ | $3$ | $(1,2,3)$ | $-1$ |
$30$ | $4$ | $(1,2,3,4)$ | $-1$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |