Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(311\!\cdots\!057\)\(\medspace = 3^{5} \cdot 16653619^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.49960857.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 14T46 |
| Parity: | even |
| Determinant: | 1.49960857.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.49960857.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 7x^{5} + 13x^{3} - 5x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 40 a + 64 + \left(20 a + 43\right)\cdot 73 + \left(66 a + 49\right)\cdot 73^{2} + \left(62 a + 31\right)\cdot 73^{3} + \left(20 a + 32\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 51 a + 26 + \left(2 a + 56\right)\cdot 73 + \left(53 a + 37\right)\cdot 73^{2} + \left(52 a + 21\right)\cdot 73^{3} + \left(48 a + 57\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 47 a + 1 + \left(6 a + 58\right)\cdot 73 + \left(30 a + 39\right)\cdot 73^{2} + \left(22 a + 40\right)\cdot 73^{3} + \left(49 a + 21\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 a + 33 + \left(70 a + 13\right)\cdot 73 + \left(19 a + 48\right)\cdot 73^{2} + \left(20 a + 53\right)\cdot 73^{3} + \left(24 a + 4\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 33 a + 38 + \left(52 a + 65\right)\cdot 73 + \left(6 a + 8\right)\cdot 73^{2} + \left(10 a + 8\right)\cdot 73^{3} + \left(52 a + 32\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 61 + 23\cdot 73 + 57\cdot 73^{2} + 58\cdot 73^{3} + 69\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 26 a + 69 + \left(66 a + 30\right)\cdot 73 + \left(42 a + 50\right)\cdot 73^{2} + \left(50 a + 4\right)\cdot 73^{3} + \left(23 a + 1\right)\cdot 73^{4} +O(73^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $6$ | ✓ |
| $21$ | $2$ | $(1,2)$ | $-4$ | |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ | |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ | |
| $70$ | $3$ | $(1,2,3)$ | $3$ | |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ | |
| $210$ | $4$ | $(1,2,3,4)$ | $-2$ | |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ | |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ | |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ | |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ | |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ | |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ | |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ | |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |