Basic invariants
| Dimension: | $10$ |
| Group: | $A_7$ |
| Conductor: | \(9115812039001375232\)\(\medspace = 2^{9} \cdot 7^{6} \cdot 73^{6} \) |
| Artin stem field: | Galois closure of 7.3.16711744.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 70 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_7$ |
| Projective stem field: | Galois closure of 7.3.16711744.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + 2x^{4} - 2x^{3} - 2x^{2} + 2x + 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 227 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 227 }$:
\( x^{2} + 220x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 128 + 168\cdot 227 + 137\cdot 227^{2} + 129\cdot 227^{3} + 178\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 55 + 149\cdot 227 + 170\cdot 227^{2} + 56\cdot 227^{3} + 186\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 157\cdot 227 + 55\cdot 227^{2} + 95\cdot 227^{3} + 2\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 77 a + 157 + \left(121 a + 116\right)\cdot 227 + \left(167 a + 170\right)\cdot 227^{2} + \left(3 a + 69\right)\cdot 227^{3} + \left(33 a + 29\right)\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 150 a + 15 + \left(105 a + 208\right)\cdot 227 + \left(59 a + 86\right)\cdot 227^{2} + \left(223 a + 155\right)\cdot 227^{3} + \left(193 a + 29\right)\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 190 a + 59 + \left(137 a + 7\right)\cdot 227 + \left(131 a + 92\right)\cdot 227^{2} + \left(194 a + 39\right)\cdot 227^{3} + \left(133 a + 210\right)\cdot 227^{4} +O(227^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 37 a + 27 + \left(89 a + 101\right)\cdot 227 + \left(95 a + 194\right)\cdot 227^{2} + \left(32 a + 134\right)\cdot 227^{3} + \left(93 a + 44\right)\cdot 227^{4} +O(227^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $70$ | $3$ | $(1,2,3)$ | $1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $360$ | $7$ | $(1,2,3,4,5,6,7)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $360$ | $7$ | $(1,3,4,5,6,7,2)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.