Basic invariants
| Dimension: | $10$ |
| Group: | $A_7$ |
| Conductor: | \(14117306610774528\)\(\medspace = 2^{9} \cdot 3^{14} \cdot 7^{8} \) |
| Artin stem field: | Galois closure of 7.3.112021056.1 |
| 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 7.3.112021056.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} + 3x^{5} + 3x^{4} - 9x^{3} + 3x^{2} + x - 3 \)
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The roots of $f$ are computed in an extension of $\Q_{ 659 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 659 }$:
\( x^{2} + 655x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 84 a + 206 + \left(212 a + 614\right)\cdot 659 + \left(642 a + 490\right)\cdot 659^{2} + \left(642 a + 529\right)\cdot 659^{3} + \left(484 a + 480\right)\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 385 + 268\cdot 659 + 129\cdot 659^{2} + 397\cdot 659^{3} + 485\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 158 + 292\cdot 659 + 202\cdot 659^{2} + 538\cdot 659^{3} + 347\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 575 a + 542 + \left(446 a + 60\right)\cdot 659 + \left(16 a + 212\right)\cdot 659^{2} + \left(16 a + 482\right)\cdot 659^{3} + \left(174 a + 459\right)\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 141 a + 604 + \left(154 a + 480\right)\cdot 659 + \left(465 a + 24\right)\cdot 659^{2} + \left(510 a + 317\right)\cdot 659^{3} + \left(575 a + 322\right)\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 518 a + 509 + \left(504 a + 297\right)\cdot 659 + \left(193 a + 413\right)\cdot 659^{2} + \left(148 a + 576\right)\cdot 659^{3} + \left(83 a + 137\right)\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 235 + 621\cdot 659 + 503\cdot 659^{2} + 453\cdot 659^{3} + 401\cdot 659^{4} +O(659^{5})\)
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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 |
| $1$ | $1$ | $()$ | $10$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $70$ | $3$ | $(1,2,3)$ | $1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $360$ | $7$ | $(1,2,3,4,5,6,7)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $360$ | $7$ | $(1,3,4,5,6,7,2)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.