Properties

 Label 6.330...971.20t30.a.a Dimension $6$ Group $S_5$ Conductor $3.302\times 10^{12}$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $6$ Group: $S_5$ Conductor: $$3301958349971$$$$\medspace = 14891^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 5.3.14891.1 Galois orbit size: $1$ Smallest permutation container: 20T30 Parity: odd Determinant: 1.14891.2t1.a.a Projective image: $S_5$ Projective stem field: 5.3.14891.1

Defining polynomial

 $f(x)$ $=$ $$x^{5} - x^{4} - 3 x^{3} + 3 x^{2} + x - 2$$  .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16 x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$14 + 13\cdot 17 + 16\cdot 17^{2} + 3\cdot 17^{3} + 3\cdot 17^{4} +O(17^{5})$$ $r_{ 2 }$ $=$ $$8 a + 1 + \left(12 a + 15\right)\cdot 17 + \left(8 a + 16\right)\cdot 17^{2} + \left(13 a + 13\right)\cdot 17^{3} + \left(4 a + 15\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 3 }$ $=$ $$14 a + 7 + 15 a\cdot 17 + \left(a + 9\right)\cdot 17^{2} + \left(3 a + 6\right)\cdot 17^{3} + \left(13 a + 7\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 4 }$ $=$ $$9 a + 9 + \left(4 a + 2\right)\cdot 17 + \left(8 a + 13\right)\cdot 17^{2} + \left(3 a + 1\right)\cdot 17^{3} + \left(12 a + 7\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 5 }$ $=$ $$3 a + 4 + \left(a + 2\right)\cdot 17 + \left(15 a + 12\right)\cdot 17^{2} + \left(13 a + 7\right)\cdot 17^{3} + 3 a\cdot 17^{4} +O(17^{5})$$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

 Cycle notation $(1,2)$ $(1,2,3,4,5)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character value $1$ $1$ $()$ $6$ $10$ $2$ $(1,2)$ $0$ $15$ $2$ $(1,2)(3,4)$ $-2$ $20$ $3$ $(1,2,3)$ $0$ $30$ $4$ $(1,2,3,4)$ $0$ $24$ $5$ $(1,2,3,4,5)$ $1$ $20$ $6$ $(1,2,3)(4,5)$ $0$

The blue line marks the conjugacy class containing complex conjugation.