Basic invariants
| Dimension: | $15$ |
| Group: | $S_7$ |
| Conductor: | \(506\!\cdots\!901\)\(\medspace = 55078981^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.55078981.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T412 |
| Parity: | even |
| Determinant: | 1.55078981.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.55078981.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - 7x^{5} + 2x^{4} + 12x^{3} - 5x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 30 a + 35 + \left(70 a + 43\right)\cdot 73 + \left(48 a + 46\right)\cdot 73^{2} + \left(12 a + 62\right)\cdot 73^{3} + \left(47 a + 59\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a + 60 + \left(35 a + 31\right)\cdot 73 + \left(46 a + 45\right)\cdot 73^{2} + \left(48 a + 44\right)\cdot 73^{3} + \left(18 a + 50\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 + 43\cdot 73 + 66\cdot 73^{3} + 2\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 a + 52 + \left(2 a + 5\right)\cdot 73 + \left(24 a + 50\right)\cdot 73^{2} + \left(60 a + 51\right)\cdot 73^{3} + \left(25 a + 42\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 a + 61 + \left(19 a + 72\right)\cdot 73 + \left(43 a + 53\right)\cdot 73^{2} + \left(36 a + 37\right)\cdot 73^{3} + \left(69 a + 62\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 69 a + 72 + \left(37 a + 59\right)\cdot 73 + \left(26 a + 3\right)\cdot 73^{2} + \left(24 a + 71\right)\cdot 73^{3} + \left(54 a + 57\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 50 a + 57 + \left(53 a + 34\right)\cdot 73 + \left(29 a + 18\right)\cdot 73^{2} + \left(36 a + 31\right)\cdot 73^{3} + \left(3 a + 15\right)\cdot 73^{4} +O(73^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.