Basic invariants
Dimension: | $20$ |
Group: | $S_7$ |
Conductor: | \(560\!\cdots\!104\)\(\medspace = 2^{62} \cdot 3^{40} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.60466176.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 70 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.60466176.1 |
Defining polynomial
$f(x)$ | $=$ |
\( x^{7} - 2x^{6} + 3x^{5} + x^{4} - 5x^{3} + 12x^{2} - 7x + 5 \)
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The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{3} + x + 14 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 3 + 8\cdot 17 + 14\cdot 17^{2} + 17^{3} + 11\cdot 17^{4} +O(17^{5})\)
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$r_{ 2 }$ | $=$ |
\( 16 a^{2} + 15 + \left(14 a^{2} + 11 a + 7\right)\cdot 17 + \left(3 a^{2} + 15 a + 16\right)\cdot 17^{2} + \left(2 a^{2} + 2\right)\cdot 17^{3} + \left(14 a^{2} + 16 a + 5\right)\cdot 17^{4} +O(17^{5})\)
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$r_{ 3 }$ | $=$ |
\( 12 a^{2} + 7 a + 9 + \left(3 a^{2} + 10 a + 7\right)\cdot 17 + \left(15 a^{2} + 7 a + 8\right)\cdot 17^{2} + \left(4 a^{2} + 12 a + 12\right)\cdot 17^{3} + \left(16 a^{2} + 15 a + 5\right)\cdot 17^{4} +O(17^{5})\)
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$r_{ 4 }$ | $=$ |
\( 10 a^{2} + 11 a + 2 + \left(15 a^{2} + 7 a + 4\right)\cdot 17 + \left(2 a^{2} + 3 a\right)\cdot 17^{2} + \left(13 a^{2} + 15 a + 1\right)\cdot 17^{3} + \left(11 a^{2} + 14\right)\cdot 17^{4} +O(17^{5})\)
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$r_{ 5 }$ | $=$ |
\( 8 a^{2} + 7 a + 4 + \left(15 a^{2} + 11 a + 8\right)\cdot 17 + \left(7 a^{2} + 2 a + 13\right)\cdot 17^{2} + \left(7 a^{2} + 16 a\right)\cdot 17^{3} + \left(14 a^{2} + 11\right)\cdot 17^{4} +O(17^{5})\)
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$r_{ 6 }$ | $=$ |
\( 10 a^{2} + 10 a + 11 + \left(3 a^{2} + 11 a + 11\right)\cdot 17 + \left(5 a^{2} + 15 a + 11\right)\cdot 17^{2} + \left(7 a^{2} + 16 a\right)\cdot 17^{3} + \left(5 a^{2} + 16 a + 5\right)\cdot 17^{4} +O(17^{5})\)
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$r_{ 7 }$ | $=$ |
\( 12 a^{2} + 16 a + 9 + \left(14 a^{2} + 15 a + 3\right)\cdot 17 + \left(15 a^{2} + 5 a + 3\right)\cdot 17^{2} + \left(15 a^{2} + 6 a + 14\right)\cdot 17^{3} + \left(5 a^{2} + 15\right)\cdot 17^{4} +O(17^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character value |
$1$ | $1$ | $()$ | $20$ |
$21$ | $2$ | $(1,2)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
$70$ | $3$ | $(1,2,3)$ | $2$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$210$ | $4$ | $(1,2,3,4)$ | $0$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.