Basic invariants
Dimension: | $2$ |
Group: | $D_5\times C_5$ |
Conductor: | \(2500\)\(\medspace = 2^{2} \cdot 5^{4} \) |
Artin stem field: | Galois closure of 10.0.400000000.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $D_5\times C_5$ |
Parity: | odd |
Determinant: | 1.100.10t1.a.c |
Projective image: | $D_5$ |
Projective stem field: | Galois closure of 5.1.6250000.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - 4x^{9} + 9x^{8} - 14x^{7} + 15x^{6} - 10x^{5} + 3x^{4} + 2x^{3} - 2x^{2} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{5} + 4x + 11 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 a^{4} + 7 a^{3} + 10 a^{2} + 4 a + 1 + \left(a^{4} + 10 a^{3} + 10 a^{2} + 10 a + 1\right)\cdot 13 + \left(4 a^{4} + 3 a^{3} + 2 a^{2} + 7 a + 3\right)\cdot 13^{2} + \left(a^{4} + 7 a^{3} + 10 a^{2} + 12 a + 4\right)\cdot 13^{3} + \left(5 a^{3} + 2 a + 1\right)\cdot 13^{4} + \left(9 a^{3} + 5 a^{2} + 3 a + 1\right)\cdot 13^{5} + \left(5 a^{4} + a^{3} + 8 a^{2} + 11\right)\cdot 13^{6} + \left(11 a^{4} + 3 a^{3} + 3 a^{2} + 5 a + 7\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 2 }$ | $=$ | \( 4 a^{4} + 7 a^{3} + 7 a^{2} + 8 a + 7 + \left(2 a^{4} + 9 a^{3} + 6 a^{2} + 11 a + 11\right)\cdot 13 + \left(11 a^{4} + 9 a^{3} + 12 a^{2} + 9 a + 6\right)\cdot 13^{2} + \left(12 a^{4} + 4 a^{3} + 3 a^{2} + 5 a + 12\right)\cdot 13^{3} + \left(7 a^{4} + a^{3} + 5 a^{2} + 8 a + 8\right)\cdot 13^{4} + \left(9 a^{4} + 9 a^{3} + 11 a^{2} + 4 a + 3\right)\cdot 13^{5} + \left(7 a^{4} + 2 a^{3} + 3 a^{2} + 3 a + 6\right)\cdot 13^{6} + \left(12 a^{4} + 11 a^{3} + 11 a^{2} + 11 a + 6\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 3 }$ | $=$ | \( 6 a^{4} + 9 a^{3} + 10 a + 3 + \left(11 a^{4} + 2 a^{3} + 10 a^{2} + 3 a + 12\right)\cdot 13 + \left(11 a^{4} + 5 a^{3} + 6 a^{2} + 7 a + 3\right)\cdot 13^{2} + \left(6 a^{4} + 2 a^{3} + 12 a^{2} + 8 a + 1\right)\cdot 13^{3} + \left(10 a^{4} + 9 a^{3} + 11 a^{2} + 12 a + 4\right)\cdot 13^{4} + \left(10 a^{4} + 6 a^{3} + 10 a^{2} + 4 a + 2\right)\cdot 13^{5} + \left(6 a^{4} + 11 a^{3} + a^{2} + a + 11\right)\cdot 13^{6} + \left(a^{4} + 6 a^{3} + 3 a^{2} + 7 a + 4\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 4 }$ | $=$ | \( 6 a^{4} + a^{3} + a^{2} + 7 a + 5 + \left(8 a^{2} + 8 a + 10\right)\cdot 13 + \left(12 a^{4} + 9 a^{3} + 5 a^{2} + 11 a + 12\right)\cdot 13^{2} + \left(11 a^{3} + 2 a^{2} + 8 a + 2\right)\cdot 13^{3} + \left(7 a^{4} + 7 a^{3} + 8 a^{2} + 8 a\right)\cdot 13^{4} + \left(5 a^{4} + 11 a^{3} + 6 a^{2} + 11\right)\cdot 13^{5} + \left(a^{4} + 6 a^{3} + 7 a^{2} + 6 a + 4\right)\cdot 13^{6} + \left(12 a^{4} + 12 a^{3} + 4 a^{2} + 9 a + 2\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 5 }$ | $=$ | \( 3 a^{4} + 5 a^{3} + 10 a + 11 + \left(9 a^{4} + 9 a^{3} + 11 a^{2} + 12 a + 9\right)\cdot 13 + \left(5 a^{3} + 4 a^{2} + 12 a + 7\right)\cdot 13^{2} + \left(9 a^{4} + 6 a^{3} + 7 a^{2} + 5\right)\cdot 13^{3} + \left(9 a^{4} + 10 a^{3} + 7 a^{2} + a + 3\right)\cdot 13^{4} + \left(6 a^{4} + 4 a^{3} + 5 a^{2} + 2 a + 12\right)\cdot 13^{5} + \left(8 a^{4} + 8 a^{3} + 2 a^{2} + a + 11\right)\cdot 13^{6} + \left(6 a^{4} + 12 a^{2} + 5 a + 10\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 6 }$ | $=$ | \( 11 a^{4} + 11 a^{3} + 2 a^{2} + 6 a + 8 + \left(8 a^{4} + 9 a^{3} + a^{2} + 8 a + 8\right)\cdot 13 + \left(4 a^{4} + 3 a^{3} + 11 a^{2} + 12\right)\cdot 13^{2} + \left(12 a^{4} + a^{3} + 8 a^{2} + 10 a + 10\right)\cdot 13^{3} + \left(3 a^{4} + 3 a^{3} + 3 a^{2} + 9 a + 5\right)\cdot 13^{4} + \left(7 a^{4} + 5 a^{3} + 6 a^{2} + 4 a + 11\right)\cdot 13^{5} + \left(2 a^{4} + 11 a^{3} + 6 a^{2} + 7 a + 5\right)\cdot 13^{6} + \left(a^{4} + 3 a^{3} + 7 a^{2} + 6 a + 6\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 7 }$ | $=$ | \( 11 a^{4} + 2 a^{3} + 12 a + 8 + \left(5 a^{4} + 9 a^{3} + 8 a^{2} + 11 a + 1\right)\cdot 13 + \left(4 a^{4} + 3 a^{3} + a^{2} + 5 a + 4\right)\cdot 13^{2} + \left(2 a^{4} + 12 a^{3} + 10 a^{2} + 6 a + 2\right)\cdot 13^{3} + \left(5 a^{4} + 11 a^{3} + 5 a^{2} + 3 a + 7\right)\cdot 13^{4} + \left(6 a^{4} + 7 a^{3} + 2 a^{2} + 2 a + 8\right)\cdot 13^{5} + \left(8 a^{4} + 10 a^{3} + a^{2} + 11 a + 6\right)\cdot 13^{6} + \left(7 a^{4} + 5 a^{3} + 11 a^{2} + 12 a + 11\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 8 }$ | $=$ | \( 3 a^{4} + a^{3} + 7 a^{2} + 11 a + 9 + \left(3 a^{4} + 12 a^{3} + 5 a^{2} + 8 a + 6\right)\cdot 13 + \left(9 a^{4} + 10 a^{3} + 3 a^{2} + 8\right)\cdot 13^{2} + \left(3 a^{3} + 3 a + 12\right)\cdot 13^{3} + \left(12 a^{4} + 5 a^{2} + a + 8\right)\cdot 13^{4} + \left(a^{4} + 12 a^{3} + 7\right)\cdot 13^{5} + \left(2 a^{4} + 2 a^{3} + 9 a^{2} + 12 a + 6\right)\cdot 13^{6} + \left(8 a^{4} + 5 a^{3} + 12 a^{2} + 10 a + 10\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 9 }$ | $=$ | \( 3 a^{3} + a + 2 + \left(3 a^{4} + 5 a^{3} + 7 a^{2} + 3 a + 11\right)\cdot 13 + \left(7 a^{4} + 11 a^{3} + 9 a^{2} + 2 a + 9\right)\cdot 13^{2} + \left(6 a^{4} + 2 a^{3} + a^{2} + 2\right)\cdot 13^{3} + \left(10 a^{4} + 3 a^{2} + 12 a + 9\right)\cdot 13^{4} + \left(3 a^{4} + 2 a^{3} + 10 a^{2} + 3 a\right)\cdot 13^{5} + \left(4 a^{4} + 9 a^{3} + 10 a^{2} + 7 a + 3\right)\cdot 13^{6} + \left(12 a^{3} + 3 a^{2} + 7 a + 6\right)\cdot 13^{7} +O(13^{8})\) |
$r_{ 10 }$ | $=$ | \( 6 a^{3} + 12 a^{2} + 9 a + 2 + \left(6 a^{4} + 9 a^{3} + 9 a^{2} + 11 a + 5\right)\cdot 13 + \left(12 a^{4} + a^{3} + 6 a^{2} + 5 a + 8\right)\cdot 13^{2} + \left(11 a^{4} + 12 a^{3} + 7 a^{2} + 8 a + 9\right)\cdot 13^{3} + \left(10 a^{4} + a^{3} + 4 a + 2\right)\cdot 13^{4} + \left(12 a^{4} + 9 a^{3} + 6 a^{2} + 12 a + 6\right)\cdot 13^{5} + \left(4 a^{4} + 12 a^{3} + a + 10\right)\cdot 13^{6} + \left(3 a^{4} + 2 a^{3} + 8 a^{2} + 2 a + 10\right)\cdot 13^{7} +O(13^{8})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 10 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 10 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$5$ | $2$ | $(1,2)(3,6)(4,10)(5,9)(7,8)$ | $0$ |
$1$ | $5$ | $(1,7,6,4,5)(2,8,3,10,9)$ | $2 \zeta_{5}^{3}$ |
$1$ | $5$ | $(1,6,5,7,4)(2,3,9,8,10)$ | $2 \zeta_{5}$ |
$1$ | $5$ | $(1,4,7,5,6)(2,10,8,9,3)$ | $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ |
$1$ | $5$ | $(1,5,4,6,7)(2,9,10,3,8)$ | $2 \zeta_{5}^{2}$ |
$2$ | $5$ | $(1,7,6,4,5)(2,9,10,3,8)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$2$ | $5$ | $(1,6,5,7,4)(2,10,8,9,3)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
$2$ | $5$ | $(2,8,3,10,9)$ | $\zeta_{5}^{2} + \zeta_{5}$ |
$2$ | $5$ | $(2,3,9,8,10)$ | $-\zeta_{5}^{3} - \zeta_{5} - 1$ |
$2$ | $5$ | $(2,10,8,9,3)$ | $\zeta_{5}^{3} + \zeta_{5}$ |
$2$ | $5$ | $(2,9,10,3,8)$ | $-\zeta_{5}^{2} - \zeta_{5} - 1$ |
$2$ | $5$ | $(1,5,4,6,7)(2,3,9,8,10)$ | $\zeta_{5}^{3} + 1$ |
$2$ | $5$ | $(1,4,7,5,6)(2,9,10,3,8)$ | $\zeta_{5} + 1$ |
$2$ | $5$ | $(1,6,5,7,4)(2,8,3,10,9)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$ |
$2$ | $5$ | $(1,7,6,4,5)(2,10,8,9,3)$ | $\zeta_{5}^{2} + 1$ |
$5$ | $10$ | $(1,8,7,3,6,10,4,9,5,2)$ | $0$ |
$5$ | $10$ | $(1,3,4,2,7,10,5,8,6,9)$ | $0$ |
$5$ | $10$ | $(1,9,6,8,5,10,7,2,4,3)$ | $0$ |
$5$ | $10$ | $(1,2,5,9,4,10,6,3,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.