Basic invariants
Dimension: | $2$ |
Group: | $D_5$ |
Conductor: | \(401\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 10.10.10368641602001.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{5}$ |
Parity: | even |
Determinant: | 1.401.2t1.a.a |
Projective image: | $D_5$ |
Projective field: | Galois closure of 10.10.10368641602001.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{10} - 2x^{9} - 20x^{8} + 2x^{7} + 69x^{6} - x^{5} - 69x^{4} + 2x^{3} + 20x^{2} - 2x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{5} + 10x^{2} + 9 \)
Roots:
$r_{ 1 }$ | $=$ | \( 9 a^{4} + 8 a^{3} + 5 a + 2 + \left(5 a^{4} + 6 a^{3} + 2 a^{2} + 3 a + 5\right)\cdot 11 + \left(7 a^{4} + 5 a^{3} + 5 a^{2} + 5 a + 9\right)\cdot 11^{2} + \left(6 a^{4} + 9 a^{3} + 6 a^{2} + 8 a + 10\right)\cdot 11^{3} + \left(a^{4} + 5 a^{3} + 9 a + 10\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 2 }$ | $=$ | \( 5 a^{4} + 5 a^{3} + 7 a^{2} + 3 a + 6 + \left(2 a^{4} + 7 a^{3} + 2 a^{2} + a + 9\right)\cdot 11 + \left(10 a^{4} + 6 a^{3} + 3 a^{2} + 2 a + 4\right)\cdot 11^{2} + \left(5 a^{4} + 9 a^{3} + 10 a^{2} + 10 a\right)\cdot 11^{3} + \left(10 a^{4} + 7 a^{3} + 2 a^{2} + 10 a + 1\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 3 }$ | $=$ | \( 8 a^{4} + 4 a^{3} + 7 a^{2} + 2 a + \left(7 a^{4} + 9 a^{3} + 10 a^{2} + 10\right)\cdot 11 + \left(a^{4} + 8 a^{3} + a^{2} + 5 a + 6\right)\cdot 11^{2} + \left(2 a^{4} + 4 a^{2} + 6 a + 2\right)\cdot 11^{3} + \left(3 a^{4} + 8 a^{3} + 8 a^{2} + 4 a + 2\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 4 }$ | $=$ | \( 7 a^{4} + 4 a^{3} + a^{2} + 7 a + \left(4 a^{4} + 2 a^{3} + 4 a + 1\right)\cdot 11 + \left(a^{3} + 3 a^{2} + 9 a + 5\right)\cdot 11^{2} + \left(4 a^{3} + 4 a^{2} + 9 a\right)\cdot 11^{3} + \left(9 a^{4} + 3 a^{3} + 8 a^{2} + a + 7\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 5 }$ | $=$ | \( 8 a^{4} + 6 a^{3} + 3 a^{2} + 5 a + 1 + \left(8 a^{4} + 7 a^{3} + 10 a^{2} + a + 10\right)\cdot 11 + \left(8 a^{4} + 4 a^{3} + 8 a^{2} + 9 a + 3\right)\cdot 11^{2} + \left(5 a^{4} + 5 a^{3} + 4 a^{2} + 5 a + 8\right)\cdot 11^{3} + \left(10 a^{4} + 3 a\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 6 }$ | $=$ | \( a^{4} + 10 a^{3} + 3 a + 3 + \left(7 a^{4} + 5 a^{3} + 8 a^{2} + a\right)\cdot 11 + \left(3 a^{4} + 8 a^{3} + 9 a^{2} + 9 a + 5\right)\cdot 11^{2} + \left(8 a^{4} + 3 a^{3} + 4 a^{2} + 9 a + 9\right)\cdot 11^{3} + \left(9 a^{4} + 6 a^{3} + 4 a^{2} + 8 a + 2\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 7 }$ | $=$ | \( a^{4} + 2 a^{3} + a^{2} + 9 a + 10 + \left(3 a^{4} + a^{3} + 5 a^{2} + 3 a + 4\right)\cdot 11 + \left(6 a^{4} + 9 a^{3} + 3 a^{2} + 5 a + 8\right)\cdot 11^{2} + \left(7 a^{3} + 4 a^{2} + 4 a\right)\cdot 11^{3} + \left(3 a^{4} + 2 a^{3} + 4 a^{2} + 3\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 8 }$ | $=$ | \( 8 a^{4} + 7 a^{3} + 3 a^{2} + 5 a + 7 + \left(7 a^{4} + 8 a^{3} + a^{2} + a + 5\right)\cdot 11 + \left(8 a^{4} + 8 a^{3} + 2 a^{2} + 4 a + 6\right)\cdot 11^{2} + \left(4 a^{4} + 3 a^{3} + 2 a^{2} + 9 a + 9\right)\cdot 11^{3} + \left(9 a^{4} + 9 a^{3} + 7 a + 9\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 9 }$ | $=$ | \( 6 a^{4} + a^{3} + 6 a^{2} + 4 a + 4 + \left(4 a^{4} + 5 a^{3} + 7 a^{2} + 9 a + 6\right)\cdot 11 + \left(a^{4} + 2 a^{2} + 9 a\right)\cdot 11^{2} + \left(4 a^{4} + 5 a^{3} + 5 a^{2} + 7 a + 6\right)\cdot 11^{3} + \left(8 a^{4} + 8 a^{3} + 4 a^{2} + 3 a + 4\right)\cdot 11^{4} +O(11^{5})\) |
$r_{ 10 }$ | $=$ | \( 2 a^{4} + 8 a^{3} + 5 a^{2} + a + 2 + \left(3 a^{4} + 7 a^{2} + 6 a + 2\right)\cdot 11 + \left(6 a^{4} + a^{3} + 3 a^{2} + 6 a + 4\right)\cdot 11^{2} + \left(5 a^{4} + 5 a^{3} + 8 a^{2} + 4 a + 6\right)\cdot 11^{3} + \left(2 a^{3} + 9 a^{2} + 3 a + 1\right)\cdot 11^{4} +O(11^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$5$ | $2$ | $(1,2)(3,5)(4,7)(6,9)(8,10)$ | $0$ |
$2$ | $5$ | $(1,3,4,6,8)(2,10,9,7,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$2$ | $5$ | $(1,4,8,3,6)(2,9,5,10,7)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.