Basic invariants
| Dimension: | $2$ |
| Group: | $D_{10}$ |
| Conductor: | \(3639\)\(\medspace = 3 \cdot 1213 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 10.0.526077196401123.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{10}$ |
| Parity: | odd |
| Projective image: | $D_5$ |
| Projective field: | Galois closure of 5.1.13242321.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{5} + 4x + 11 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a^{4} + 2 a^{3} + 3 a^{2} + 12 a + \left(2 a^{4} + 5 a^{3} + 5 a^{2} + 9 a + 12\right)\cdot 13 + \left(9 a^{4} + 4 a^{3} + a^{2} + 10 a + 9\right)\cdot 13^{2} + \left(6 a^{4} + 2 a^{3} + a^{2} + 12 a + 7\right)\cdot 13^{3} + \left(11 a^{3} + 5 a^{2} + 6 a\right)\cdot 13^{4} + \left(10 a^{3} + a^{2} + 3 a + 11\right)\cdot 13^{5} + \left(3 a^{3} + 9 a^{2} + 11 a\right)\cdot 13^{6} + \left(9 a^{4} + 6 a^{3} + 10 a^{2} + a + 1\right)\cdot 13^{7} + \left(6 a^{3} + 10 a^{2} + 3 a + 9\right)\cdot 13^{8} + \left(5 a^{4} + 11 a^{3} + 4 a^{2} + 8 a + 12\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 2 }$ | $=$ |
\( 3 a^{4} + 5 a^{3} + 5 a^{2} + 2 a + 9 + \left(3 a^{4} + 7 a^{3} + 8 a^{2} + 11 a + 10\right)\cdot 13 + \left(11 a^{4} + 2 a^{3} + 8 a^{2} + 2 a\right)\cdot 13^{2} + \left(11 a^{4} + 8 a^{3} + 12 a^{2} + a + 10\right)\cdot 13^{3} + \left(a^{4} + 3 a^{3} + a^{2} + 11 a + 9\right)\cdot 13^{4} + \left(11 a^{4} + 3 a + 11\right)\cdot 13^{5} + \left(8 a^{4} + a^{3} + 11 a^{2} + 3 a + 11\right)\cdot 13^{6} + \left(5 a^{4} + 8 a^{3} + 7 a^{2} + 12 a + 6\right)\cdot 13^{7} + \left(11 a^{4} + 8 a^{3} + 2 a^{2} + 11\right)\cdot 13^{8} + \left(2 a^{4} + 3 a^{3} + 4 a^{2} + 5 a + 4\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 3 }$ | $=$ |
\( 7 a^{4} + 2 a^{3} + 3 a^{2} + 7 a + 5 + \left(6 a^{4} + 9 a^{3} + 3 a^{2} + 2 a + 2\right)\cdot 13 + \left(2 a^{4} + 8 a^{3} + 8 a^{2} + 6 a + 4\right)\cdot 13^{2} + \left(7 a^{4} + 8 a^{3} + 6 a^{2} + 7 a + 9\right)\cdot 13^{3} + \left(12 a^{4} + 3 a^{3} + 3 a^{2} + 2\right)\cdot 13^{4} + \left(a^{4} + 7 a^{3} + 5 a^{2} + a + 4\right)\cdot 13^{5} + \left(11 a^{4} + 3 a^{3} + 11 a^{2} + 8 a + 5\right)\cdot 13^{6} + \left(6 a^{3} + 4 a^{2} + 11 a + 11\right)\cdot 13^{7} + \left(3 a^{4} + 7 a^{3} + 7 a^{2} + a + 8\right)\cdot 13^{8} + \left(5 a^{4} + 3 a^{3} + 2 a^{2} + 5 a + 10\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 4 }$ | $=$ |
\( 7 a^{4} + 2 a^{3} + 11 a^{2} + 12 a + 5 + \left(8 a^{4} + 6 a^{3} + 12 a^{2} + 8 a + 11\right)\cdot 13 + \left(10 a^{4} + 7 a^{3} + 11 a^{2} + 2 a + 6\right)\cdot 13^{2} + \left(5 a^{4} + 2 a^{3} + 7 a^{2} + 7\right)\cdot 13^{3} + \left(7 a^{4} + 7 a^{3} + 5 a^{2} + 2 a + 4\right)\cdot 13^{4} + \left(11 a^{4} + a^{3} + 3 a^{2} + 4 a + 6\right)\cdot 13^{5} + \left(a^{4} + 10 a^{3} + 4 a^{2} + a + 9\right)\cdot 13^{6} + \left(2 a^{4} + 2 a^{3} + 6 a^{2} + 3 a + 7\right)\cdot 13^{7} + \left(2 a^{4} + 6 a^{3} + 7 a^{2} + 11 a + 8\right)\cdot 13^{8} + \left(12 a^{4} + 8 a^{3} + 3 a^{2} + 12 a + 1\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 5 }$ | $=$ |
\( 9 a^{4} + 2 a^{3} + 2 a^{2} + 6 a + 1 + \left(3 a^{4} + 12 a^{3} + 2 a^{2} + 3 a + 1\right)\cdot 13 + \left(9 a^{4} + a^{3} + 5 a^{2} + 5\right)\cdot 13^{2} + \left(10 a^{4} + 12 a^{3} + 7 a^{2} + 3 a + 2\right)\cdot 13^{3} + \left(6 a^{4} + 9 a^{3} + 8 a^{2} + 9 a + 5\right)\cdot 13^{4} + \left(8 a^{4} + 5 a^{3} + 10 a^{2} + a + 4\right)\cdot 13^{5} + \left(11 a^{3} + 10 a^{2} + 11 a\right)\cdot 13^{6} + \left(4 a^{4} + 12 a^{3} + 10 a^{2} + 2 a + 6\right)\cdot 13^{7} + \left(5 a^{4} + 10 a^{3} + 9 a^{2} + 6 a\right)\cdot 13^{8} + \left(4 a^{4} + 2 a^{3} + 7 a^{2} + 8\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 6 }$ | $=$ |
\( 6 a^{3} + 11 a^{2} + 5 a + 2 + \left(6 a^{4} + 10 a^{3} + 3 a^{2} + 12 a + 9\right)\cdot 13 + \left(6 a^{4} + 2 a^{3} + 11 a^{2} + 2 a + 3\right)\cdot 13^{2} + \left(10 a^{4} + 7 a^{3} + 9 a^{2} + 2 a + 8\right)\cdot 13^{3} + \left(3 a^{4} + 11 a^{3} + 4 a^{2} + 10\right)\cdot 13^{4} + \left(2 a^{4} + a^{3} + 5 a^{2} + 11 a + 6\right)\cdot 13^{5} + \left(11 a^{4} + 8 a^{3} + 11 a^{2} + 6 a + 11\right)\cdot 13^{6} + \left(4 a^{4} + 11 a^{3} + a^{2} + 6 a + 6\right)\cdot 13^{7} + \left(3 a^{4} + 6 a^{3} + 5 a^{2} + 7 a + 6\right)\cdot 13^{8} + \left(10 a^{4} + 9 a^{2} + 2 a + 2\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 7 }$ | $=$ |
\( a^{4} + 5 a^{3} + a^{2} + 4 a + \left(10 a^{4} + 9 a^{3} + 11 a^{2} + 10 a + 4\right)\cdot 13 + \left(11 a^{4} + 4 a^{3} + 9 a^{2} + 9 a + 10\right)\cdot 13^{2} + \left(10 a^{3} + 12 a^{2} + 8\right)\cdot 13^{3} + \left(10 a^{4} + 11 a^{3} + 7 a^{2} + 9 a + 4\right)\cdot 13^{4} + \left(5 a^{4} + 12 a^{3} + 6 a^{2} + 5 a + 2\right)\cdot 13^{5} + \left(11 a^{4} + 5 a^{2} + 5 a + 7\right)\cdot 13^{6} + \left(2 a^{4} + 12 a^{3} + 6 a^{2} + a\right)\cdot 13^{7} + \left(6 a^{4} + 10 a^{3} + 8 a^{2} + 4 a\right)\cdot 13^{8} + \left(6 a^{4} + 4 a^{3} + 9 a^{2} + 5 a + 6\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 11 a^{4} + 5 a^{3} + a^{2} + 11 a + 6 + \left(6 a^{3} + 6 a^{2} + 3 a\right)\cdot 13 + \left(6 a^{4} + 8 a^{3} + 4 a^{2} + 9 a + 5\right)\cdot 13^{2} + \left(10 a^{4} + 4 a^{3} + 5 a^{2} + 5 a\right)\cdot 13^{3} + \left(7 a^{4} + a^{3} + 8 a^{2} + a\right)\cdot 13^{4} + \left(a^{4} + 5 a^{3} + 5 a^{2} + 2\right)\cdot 13^{5} + \left(8 a^{4} + 4 a^{3} + 12 a^{2} + 4 a + 12\right)\cdot 13^{6} + \left(3 a^{4} + 9 a^{3} + 2 a^{2} + 10 a + 2\right)\cdot 13^{7} + \left(11 a^{4} + 8 a^{3} + 5 a^{2} + 8 a + 3\right)\cdot 13^{8} + \left(4 a^{4} + 6 a^{3} + a + 11\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 9 }$ | $=$ |
\( 5 a^{3} + 7 a^{2} + 2 a + 6 + \left(5 a^{4} + 6 a^{3} + 2 a^{2} + a + 10\right)\cdot 13 + \left(7 a^{4} + 3 a^{3} + 12 a^{2} + 6 a + 6\right)\cdot 13^{2} + \left(8 a^{4} + 2 a^{2} + 2 a + 8\right)\cdot 13^{3} + \left(11 a^{4} + 7 a^{3} + 3 a^{2} + 7 a + 7\right)\cdot 13^{4} + \left(3 a^{4} + 5 a^{2} + 2 a + 2\right)\cdot 13^{5} + \left(12 a^{4} + 10 a^{3} + 3 a^{2} + 7 a + 1\right)\cdot 13^{6} + \left(9 a^{4} + 10 a^{3} + 6 a^{2} + 6 a + 4\right)\cdot 13^{7} + \left(a^{4} + 7 a^{3} + 3 a^{2} + 3 a + 7\right)\cdot 13^{8} + \left(12 a^{4} + 12 a^{3} + 7 a^{2} + 12 a + 1\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 10 }$ | $=$ |
\( 11 a^{4} + 5 a^{3} + 8 a^{2} + 4 a + 6 + \left(5 a^{4} + 5 a^{3} + 9 a^{2} + a + 3\right)\cdot 13 + \left(3 a^{4} + 7 a^{3} + 4 a^{2} + a + 12\right)\cdot 13^{2} + \left(5 a^{4} + 8 a^{3} + 11 a^{2} + 3 a + 1\right)\cdot 13^{3} + \left(2 a^{4} + 10 a^{3} + 2 a^{2} + 4 a + 6\right)\cdot 13^{4} + \left(5 a^{4} + 5 a^{3} + 8 a^{2} + 5 a\right)\cdot 13^{5} + \left(12 a^{4} + 11 a^{3} + 11 a^{2} + 6 a + 5\right)\cdot 13^{6} + \left(8 a^{4} + 10 a^{3} + 6 a^{2} + 8 a + 4\right)\cdot 13^{7} + \left(6 a^{4} + 3 a^{3} + 4 a^{2} + 4 a + 9\right)\cdot 13^{8} + \left(a^{4} + 10 a^{3} + 2 a^{2} + 11 a + 5\right)\cdot 13^{9} +O(13^{10})\)
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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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,7)(2,5)(3,8)(4,10)(6,9)$ | $-2$ | $-2$ |
| $5$ | $2$ | $(1,9)(2,10)(4,5)(6,7)$ | $0$ | $0$ |
| $5$ | $2$ | $(1,6)(2,4)(3,8)(5,10)(7,9)$ | $0$ | $0$ |
| $2$ | $5$ | $(1,5,3,4,9)(2,8,10,6,7)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
| $2$ | $5$ | $(1,3,9,5,4)(2,10,7,8,6)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
| $2$ | $10$ | $(1,2,3,10,9,7,5,8,4,6)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $2$ | $10$ | $(1,10,5,6,3,7,4,2,9,8)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |