Basic invariants
Dimension: | $21$ |
Group: | $A_7$ |
Conductor: | \(481\!\cdots\!216\)\(\medspace = 2^{18} \cdot 7^{10} \cdot 73^{16} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.3.16711744.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_7$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $A_7$ |
Projective stem field: | Galois closure of 7.3.16711744.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{6} + 2x^{4} - 2x^{3} - 2x^{2} + 2x + 2 \) . |
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 |
$1$ | $1$ | $()$ | $21$ |
$105$ | $2$ | $(1,2)(3,4)$ | $1$ |
$70$ | $3$ | $(1,2,3)$ | $-3$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
$360$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$360$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.