Basic invariants
Dimension: | $14$ |
Group: | $A_7$ |
Conductor: | \(247\!\cdots\!056\)\(\medspace = 2^{12} \cdot 3^{24} \cdot 11^{8} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 7.3.50808384.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_7$ |
Parity: | even |
Projective image: | $A_7$ |
Projective field: | Galois closure of 7.3.50808384.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$:
\( x^{2} + 102x + 5 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 21 a + 28 + \left(22 a + 5\right)\cdot 103 + \left(42 a + 60\right)\cdot 103^{2} + \left(13 a + 62\right)\cdot 103^{3} + \left(17 a + 70\right)\cdot 103^{4} +O(103^{5})\)
$r_{ 2 }$ |
$=$ |
\( 67 + 7\cdot 103 + 16\cdot 103^{2} + 44\cdot 103^{3} + 46\cdot 103^{4} +O(103^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 84 + 21\cdot 103 + 32\cdot 103^{2} + 43\cdot 103^{3} + 59\cdot 103^{4} +O(103^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 82 a + 49 + \left(80 a + 6\right)\cdot 103 + \left(60 a + 80\right)\cdot 103^{2} + \left(89 a + 33\right)\cdot 103^{3} + \left(85 a + 74\right)\cdot 103^{4} +O(103^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 66 + 36\cdot 103 + 56\cdot 103^{2} + 11\cdot 103^{3} + 73\cdot 103^{4} +O(103^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 77 a + 73 + \left(68 a + 16\right)\cdot 103 + \left(42 a + 45\right)\cdot 103^{2} + \left(20 a + 16\right)\cdot 103^{3} + \left(58 a + 25\right)\cdot 103^{4} +O(103^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 26 a + 47 + \left(34 a + 8\right)\cdot 103 + \left(60 a + 19\right)\cdot 103^{2} + \left(82 a + 97\right)\cdot 103^{3} + \left(44 a + 62\right)\cdot 103^{4} +O(103^{5})\)
| |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $14$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $2$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
$360$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$360$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |