Properties

 Label 2.25992.6t3.b.a Dimension $2$ Group $D_{6}$ Conductor $25992$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$25992$$$$\medspace = 2^{3} \cdot 3^{2} \cdot 19^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.0.225194688.10 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Determinant: 1.8.2t1.b.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.2888.1

Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} - 12x^{4} - 27x^{3} + 176x^{2} + 91x + 49$$ x^6 - x^5 - 12*x^4 - 27*x^3 + 176*x^2 + 91*x + 49 .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{2} + 12x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$2 + 12\cdot 13 + 11\cdot 13^{2} + 3\cdot 13^{3} + 9\cdot 13^{4} + 11\cdot 13^{5} + 10\cdot 13^{6} + 9\cdot 13^{7} + 8\cdot 13^{8} +O(13^{9})$$ 2 + 12*13 + 11*13^2 + 3*13^3 + 9*13^4 + 11*13^5 + 10*13^6 + 9*13^7 + 8*13^8+O(13^9) $r_{ 2 }$ $=$ $$2 a + 3 + 2\cdot 13 + \left(3 a + 2\right)\cdot 13^{2} + \left(9 a + 3\right)\cdot 13^{3} + \left(3 a + 7\right)\cdot 13^{4} + \left(4 a + 5\right)\cdot 13^{5} + \left(5 a + 11\right)\cdot 13^{6} + \left(7 a + 4\right)\cdot 13^{7} + \left(12 a + 8\right)\cdot 13^{8} +O(13^{9})$$ 2*a + 3 + 2*13 + (3*a + 2)*13^2 + (9*a + 3)*13^3 + (3*a + 7)*13^4 + (4*a + 5)*13^5 + (5*a + 11)*13^6 + (7*a + 4)*13^7 + (12*a + 8)*13^8+O(13^9) $r_{ 3 }$ $=$ $$6 + 6\cdot 13 + 12\cdot 13^{2} + 12\cdot 13^{3} + 5\cdot 13^{4} + 6\cdot 13^{5} + 4\cdot 13^{6} + 8\cdot 13^{7} + 9\cdot 13^{8} +O(13^{9})$$ 6 + 6*13 + 12*13^2 + 12*13^3 + 5*13^4 + 6*13^5 + 4*13^6 + 8*13^7 + 9*13^8+O(13^9) $r_{ 4 }$ $=$ $$11 a + 5 + 12 a\cdot 13 + \left(9 a + 5\right)\cdot 13^{2} + \left(3 a + 9\right)\cdot 13^{3} + \left(9 a + 1\right)\cdot 13^{4} + \left(8 a + 6\right)\cdot 13^{5} + \left(7 a + 12\right)\cdot 13^{6} + \left(5 a + 6\right)\cdot 13^{7} +O(13^{9})$$ 11*a + 5 + 12*a*13 + (9*a + 5)*13^2 + (3*a + 9)*13^3 + (9*a + 1)*13^4 + (8*a + 6)*13^5 + (7*a + 12)*13^6 + (5*a + 6)*13^7+O(13^9) $r_{ 5 }$ $=$ $$6 a + 9 + 9 a\cdot 13 + \left(9 a + 10\right)\cdot 13^{2} + \left(a + 8\right)\cdot 13^{3} + \left(3 a + 6\right)\cdot 13^{4} + 11 a\cdot 13^{5} + \left(2 a + 4\right)\cdot 13^{6} + 11 a\cdot 13^{7} + \left(2 a + 10\right)\cdot 13^{8} +O(13^{9})$$ 6*a + 9 + 9*a*13 + (9*a + 10)*13^2 + (a + 8)*13^3 + (3*a + 6)*13^4 + 11*a*13^5 + (2*a + 4)*13^6 + 11*a*13^7 + (2*a + 10)*13^8+O(13^9) $r_{ 6 }$ $=$ $$7 a + 2 + \left(3 a + 4\right)\cdot 13 + \left(3 a + 10\right)\cdot 13^{2} + 11 a\cdot 13^{3} + \left(9 a + 8\right)\cdot 13^{4} + \left(a + 8\right)\cdot 13^{5} + \left(10 a + 8\right)\cdot 13^{6} + \left(a + 8\right)\cdot 13^{7} + \left(10 a + 1\right)\cdot 13^{8} +O(13^{9})$$ 7*a + 2 + (3*a + 4)*13 + (3*a + 10)*13^2 + 11*a*13^3 + (9*a + 8)*13^4 + (a + 8)*13^5 + (10*a + 8)*13^6 + (a + 8)*13^7 + (10*a + 1)*13^8+O(13^9)

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,3)(2,5)(4,6)$ $(1,2,4)(3,5,6)$ $(2,4)(5,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.