Basic invariants
| Dimension: | $21$ |
| Group: | $S_7$ |
| Conductor: | \(109\!\cdots\!601\)\(\medspace = 7^{16} \cdot 8951^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.438599.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 84 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.438599.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + 2x^{5} - 2x^{4} + x^{3} + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{2} + 78x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 74 a + 6 + \left(24 a + 68\right)\cdot 79 + \left(20 a + 74\right)\cdot 79^{2} + \left(16 a + 31\right)\cdot 79^{3} + \left(38 a + 59\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 + 12\cdot 79 + 23\cdot 79^{2} + 18\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 1 + \left(54 a + 19\right)\cdot 79 + \left(58 a + 70\right)\cdot 79^{2} + \left(62 a + 27\right)\cdot 79^{3} + \left(40 a + 2\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 76 a + 44 + \left(68 a + 31\right)\cdot 79 + \left(41 a + 19\right)\cdot 79^{2} + \left(56 a + 43\right)\cdot 79^{3} + \left(a + 62\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 74 + 41\cdot 79 + 30\cdot 79^{2} + 2\cdot 79^{3} + 73\cdot 79^{4} +O(79^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 46 + 39\cdot 79 + 26\cdot 79^{2} + 55\cdot 79^{3} + 21\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 3 a + 41 + \left(10 a + 24\right)\cdot 79 + \left(37 a + 71\right)\cdot 79^{2} + \left(22 a + 57\right)\cdot 79^{3} + \left(77 a + 7\right)\cdot 79^{4} +O(79^{5})\)
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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 |
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.