# Properties

 Label 2.3800.6t3.b Dimension $2$ Group $D_{6}$ Conductor $3800$ Indicator $1$

# Learn more

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$3800$$$$\medspace = 2^{3} \cdot 5^{2} \cdot 19$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.2888000.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.152.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$3 a + 8 + \left(3 a + 6\right)\cdot 11 + \left(2 a + 8\right)\cdot 11^{2} + \left(2 a + 4\right)\cdot 11^{3} + \left(a + 6\right)\cdot 11^{4} + \left(8 a + 2\right)\cdot 11^{5} + 3\cdot 11^{6} + 6 a\cdot 11^{7} +O(11^{8})$$ 3*a + 8 + (3*a + 6)*11 + (2*a + 8)*11^2 + (2*a + 4)*11^3 + (a + 6)*11^4 + (8*a + 2)*11^5 + 3*11^6 + 6*a*11^7+O(11^8) $r_{ 2 }$ $=$ $$6 a + 5 + \left(4 a + 4\right)\cdot 11 + \left(3 a + 4\right)\cdot 11^{2} + \left(9 a + 4\right)\cdot 11^{3} + 10\cdot 11^{4} + \left(6 a + 10\right)\cdot 11^{5} + \left(3 a + 6\right)\cdot 11^{6} + \left(8 a + 5\right)\cdot 11^{7} +O(11^{8})$$ 6*a + 5 + (4*a + 4)*11 + (3*a + 4)*11^2 + (9*a + 4)*11^3 + 10*11^4 + (6*a + 10)*11^5 + (3*a + 6)*11^6 + (8*a + 5)*11^7+O(11^8) $r_{ 3 }$ $=$ $$9 + 11 + 10\cdot 11^{2} + 8\cdot 11^{3} + 3\cdot 11^{5} + 11^{6} + 10\cdot 11^{7} +O(11^{8})$$ 9 + 11 + 10*11^2 + 8*11^3 + 3*11^5 + 11^6 + 10*11^7+O(11^8) $r_{ 4 }$ $=$ $$7 + 8\cdot 11 + 3\cdot 11^{2} + 9\cdot 11^{3} + 11^{4} + 3\cdot 11^{5} + 8\cdot 11^{6} + 11^{7} +O(11^{8})$$ 7 + 8*11 + 3*11^2 + 9*11^3 + 11^4 + 3*11^5 + 8*11^6 + 11^7+O(11^8) $r_{ 5 }$ $=$ $$8 a + 9 + \left(7 a + 5\right)\cdot 11 + \left(8 a + 3\right)\cdot 11^{2} + 8 a\cdot 11^{3} + \left(9 a + 9\right)\cdot 11^{4} + 2 a\cdot 11^{5} + \left(10 a + 9\right)\cdot 11^{6} + \left(4 a + 1\right)\cdot 11^{7} +O(11^{8})$$ 8*a + 9 + (7*a + 5)*11 + (8*a + 3)*11^2 + 8*a*11^3 + (9*a + 9)*11^4 + 2*a*11^5 + (10*a + 9)*11^6 + (4*a + 1)*11^7+O(11^8) $r_{ 6 }$ $=$ $$5 a + 7 + \left(6 a + 5\right)\cdot 11 + \left(7 a + 2\right)\cdot 11^{2} + \left(a + 5\right)\cdot 11^{3} + \left(10 a + 4\right)\cdot 11^{4} + \left(4 a + 1\right)\cdot 11^{5} + \left(7 a + 4\right)\cdot 11^{6} + \left(2 a + 2\right)\cdot 11^{7} +O(11^{8})$$ 5*a + 7 + (6*a + 5)*11 + (7*a + 2)*11^2 + (a + 5)*11^3 + (10*a + 4)*11^4 + (4*a + 1)*11^5 + (7*a + 4)*11^6 + (2*a + 2)*11^7+O(11^8)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,2)(3,4)(5,6)$ $-2$ $3$ $2$ $(1,3)(2,4)$ $0$ $3$ $2$ $(1,4)(2,3)(5,6)$ $0$ $2$ $3$ $(1,5,3)(2,6,4)$ $-1$ $2$ $6$ $(1,6,3,2,5,4)$ $1$
The blue line marks the conjugacy class containing complex conjugation.