Basic invariants
Dimension: | $14$ |
Group: | $S_7$ |
Conductor: | \(350\!\cdots\!871\)\(\medspace = 319831^{9} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.319831.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 30T565 |
Parity: | odd |
Determinant: | 1.319831.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.319831.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + 2x^{4} - 3x^{3} + 4x^{2} - 3x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: \( x^{2} + 70x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 33 a + 28 + \left(10 a + 67\right)\cdot 73 + \left(47 a + 5\right)\cdot 73^{2} + \left(29 a + 43\right)\cdot 73^{3} + \left(28 a + 23\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 2 }$ | $=$ | \( 35 + 12\cdot 73 + 24\cdot 73^{2} + 56\cdot 73^{3} + 64\cdot 73^{4} +O(73^{5})\) |
$r_{ 3 }$ | $=$ | \( 64 a + \left(69 a + 38\right)\cdot 73 + \left(18 a + 38\right)\cdot 73^{2} + \left(24 a + 42\right)\cdot 73^{3} + \left(51 a + 63\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 4 }$ | $=$ | \( 9 a + 46 + \left(3 a + 37\right)\cdot 73 + \left(54 a + 25\right)\cdot 73^{2} + \left(48 a + 23\right)\cdot 73^{3} + \left(21 a + 47\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 5 }$ | $=$ | \( 13 + 8\cdot 73 + 41\cdot 73^{2} + 8\cdot 73^{3} + 69\cdot 73^{4} +O(73^{5})\) |
$r_{ 6 }$ | $=$ | \( 40 a + 54 + \left(62 a + 65\right)\cdot 73 + \left(25 a + 63\right)\cdot 73^{2} + \left(43 a + 11\right)\cdot 73^{3} + \left(44 a + 6\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 7 }$ | $=$ | \( 44 + 62\cdot 73 + 19\cdot 73^{2} + 33\cdot 73^{3} + 17\cdot 73^{4} +O(73^{5})\) |
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 |
$1$ | $1$ | $()$ | $14$ |
$21$ | $2$ | $(1,2)$ | $-4$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $-1$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$210$ | $4$ | $(1,2,3,4)$ | $2$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.