Basic invariants
| Dimension: | $8$ |
| Group: | $A_6$ |
| Conductor: | \(227081481823729\)\(\medspace = 13^{6} \cdot 19^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.3722098081.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_6$ |
| Parity: | even |
| Projective image: | $A_6$ |
| Projective field: | Galois closure of 6.2.3722098081.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 45 a + 16 + \left(28 a + 36\right)\cdot 73 + \left(33 a + 12\right)\cdot 73^{2} + \left(56 a + 10\right)\cdot 73^{3} + \left(67 a + 12\right)\cdot 73^{4} + \left(44 a + 22\right)\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 a + 5 + \left(44 a + 4\right)\cdot 73 + \left(39 a + 11\right)\cdot 73^{2} + 16 a\cdot 73^{3} + \left(5 a + 13\right)\cdot 73^{4} + \left(28 a + 16\right)\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 69 a + 15 + 64 a\cdot 73 + \left(42 a + 55\right)\cdot 73^{2} + \left(47 a + 10\right)\cdot 73^{3} + \left(49 a + 33\right)\cdot 73^{4} + \left(a + 55\right)\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 3 + \left(8 a + 53\right)\cdot 73 + \left(30 a + 45\right)\cdot 73^{2} + \left(25 a + 37\right)\cdot 73^{3} + \left(23 a + 61\right)\cdot 73^{4} + \left(71 a + 10\right)\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 + 34\cdot 73 + 56\cdot 73^{2} + 20\cdot 73^{3} + 57\cdot 73^{4} + 67\cdot 73^{5} +O(73^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 + 18\cdot 73 + 38\cdot 73^{2} + 66\cdot 73^{3} + 41\cdot 73^{4} + 46\cdot 73^{5} +O(73^{6})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $8$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |