Basic invariants
Dimension: | $2$ |
Group: | $D_{5}$ |
Conductor: | \(401\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 5.5.160801.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{5}$ |
Parity: | even |
Projective image: | $D_5$ |
Projective field: | Galois closure of 5.5.160801.1 |
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:
$r_{ 1 }$ | $=$ |
\( 5 a + 12 + \left(6 a + 6\right)\cdot 13 + \left(4 a + 11\right)\cdot 13^{2} + \left(11 a + 5\right)\cdot 13^{3} + \left(a + 3\right)\cdot 13^{4} +O(13^{5})\)
$r_{ 2 }$ |
$=$ |
\( 11 a + 4 + \left(3 a + 3\right)\cdot 13 + \left(12 a + 5\right)\cdot 13^{2} + 5\cdot 13^{3} + \left(6 a + 11\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 2 a + 2 + \left(9 a + 9\right)\cdot 13 + \left(12 a + 7\right)\cdot 13^{3} + \left(6 a + 3\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 5 + 11\cdot 13 + 11\cdot 13^{2} + 7\cdot 13^{3} +O(13^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 8 a + 4 + \left(6 a + 8\right)\cdot 13 + \left(8 a + 9\right)\cdot 13^{2} + \left(a + 12\right)\cdot 13^{3} + \left(11 a + 6\right)\cdot 13^{4} +O(13^{5})\)
| |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$5$ | $2$ | $(1,4)(2,5)$ | $0$ | $0$ |
$2$ | $5$ | $(1,5,2,4,3)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
$2$ | $5$ | $(1,2,3,5,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |