Basic invariants
| Dimension: | $4$ |
| Group: | $\PGL(2,5)$ |
| Conductor: | \(10731125\)\(\medspace = 5^{3} \cdot 293^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.10731125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.5.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 6.2.10731125.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} + 4x^{3} - 3x^{2} - 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{2} + 78x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 48\cdot 79 + 77\cdot 79^{2} + 35\cdot 79^{3} + 33\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 54\cdot 79 + 52\cdot 79^{2} + 10\cdot 79^{3} + 5\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 70 a + 42 + \left(51 a + 37\right)\cdot 79 + \left(29 a + 29\right)\cdot 79^{2} + \left(13 a + 69\right)\cdot 79^{3} + \left(77 a + 34\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 77 a + 70 + \left(21 a + 26\right)\cdot 79 + \left(63 a + 14\right)\cdot 79^{2} + \left(76 a + 27\right)\cdot 79^{3} + \left(64 a + 38\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 68 + \left(57 a + 50\right)\cdot 79 + \left(15 a + 55\right)\cdot 79^{2} + \left(2 a + 40\right)\cdot 79^{3} + \left(14 a + 26\right)\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 33 + \left(27 a + 19\right)\cdot 79 + \left(49 a + 7\right)\cdot 79^{2} + \left(65 a + 53\right)\cdot 79^{3} + \left(a + 19\right)\cdot 79^{4} +O(79^{5})\)
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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 value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,6)(2,5)(3,4)$ | $-2$ |
| $15$ | $2$ | $(2,6)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,6,4)(2,3,5)$ | $1$ |
| $30$ | $4$ | $(2,4,6,3)$ | $0$ |
| $24$ | $5$ | $(1,2,4,6,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,6,3,4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.