# Properties

 Label 2.2240.6t3.d Dimension $2$ Group $D_{6}$ Conductor $2240$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$2240$$$$\medspace = 2^{6} \cdot 5 \cdot 7$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.2508800.3 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: 3.1.140.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{2} + 21 x + 5$$
Roots:
 $r_{ 1 }$ $=$ $$21 a + 3 + \left(22 a + 7\right)\cdot 23 + \left(20 a + 1\right)\cdot 23^{2} + \left(4 a + 18\right)\cdot 23^{3} + \left(12 a + 12\right)\cdot 23^{4} + \left(14 a + 21\right)\cdot 23^{5} + \left(13 a + 22\right)\cdot 23^{6} +O(23^{7})$$ $r_{ 2 }$ $=$ $$2 a + 22 + 8\cdot 23 + \left(2 a + 20\right)\cdot 23^{2} + \left(18 a + 6\right)\cdot 23^{3} + \left(10 a + 9\right)\cdot 23^{4} + \left(8 a + 15\right)\cdot 23^{5} + \left(9 a + 12\right)\cdot 23^{6} +O(23^{7})$$ $r_{ 3 }$ $=$ $$20 + 22\cdot 23 + 21\cdot 23^{2} + 15\cdot 23^{3} + 18\cdot 23^{4} + 13\cdot 23^{5} + 13\cdot 23^{6} +O(23^{7})$$ $r_{ 4 }$ $=$ $$2 a + 20 + 15\cdot 23 + \left(2 a + 21\right)\cdot 23^{2} + \left(18 a + 4\right)\cdot 23^{3} + \left(10 a + 10\right)\cdot 23^{4} + \left(8 a + 1\right)\cdot 23^{5} + 9 a\cdot 23^{6} +O(23^{7})$$ $r_{ 5 }$ $=$ $$21 a + 1 + \left(22 a + 14\right)\cdot 23 + \left(20 a + 2\right)\cdot 23^{2} + \left(4 a + 16\right)\cdot 23^{3} + \left(12 a + 13\right)\cdot 23^{4} + \left(14 a + 7\right)\cdot 23^{5} + \left(13 a + 10\right)\cdot 23^{6} +O(23^{7})$$ $r_{ 6 }$ $=$ $$3 + 23^{2} + 7\cdot 23^{3} + 4\cdot 23^{4} + 9\cdot 23^{5} + 9\cdot 23^{6} +O(23^{7})$$

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

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

### Character values on conjugacy classes

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