Basic invariants
| Dimension: | $10$ |
| Group: | $A_7$ |
| Conductor: | \(965\!\cdots\!361\)\(\medspace = 149^{6} \cdot 211^{6} \) |
| Artin stem field: | Galois closure of 7.7.988410721.1 |
| 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 7.7.988410721.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} - 7x^{5} + 11x^{4} + 16x^{3} - 14x^{2} - 11x + 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 44 + 46\cdot 53 + 14\cdot 53^{2} + 25\cdot 53^{3} + 50\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 28 + \left(18 a + 26\right)\cdot 53 + \left(41 a + 19\right)\cdot 53^{2} + \left(29 a + 22\right)\cdot 53^{3} + \left(14 a + 17\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 49 + 42\cdot 53 + 16\cdot 53^{2} + 6\cdot 53^{3} + 22\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 48 a + 48 + \left(34 a + 40\right)\cdot 53 + \left(11 a + 7\right)\cdot 53^{2} + \left(23 a + 47\right)\cdot 53^{3} + \left(38 a + 45\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a + 30 + \left(13 a + 11\right)\cdot 53 + \left(26 a + 41\right)\cdot 53^{2} + \left(33 a + 29\right)\cdot 53^{3} + \left(36 a + 3\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 37 a + 41 + \left(39 a + 48\right)\cdot 53 + \left(26 a + 26\right)\cdot 53^{2} + \left(19 a + 31\right)\cdot 53^{3} + \left(16 a + 10\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 27 + 47\cdot 53 + 31\cdot 53^{2} + 49\cdot 53^{3} + 8\cdot 53^{4} +O(53^{5})\)
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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 |
| $1$ | $1$ | $()$ | $10$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $70$ | $3$ | $(1,2,3)$ | $1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $360$ | $7$ | $(1,2,3,4,5,6,7)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $360$ | $7$ | $(1,3,4,5,6,7,2)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.