# Properties

 Label 2.79.5t2.a.a Dimension $2$ Group $D_{5}$ Conductor $79$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{5}$ Conductor: $$79$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 5.1.6241.1 Galois orbit size: $2$ Smallest permutation container: $D_{5}$ Parity: odd Determinant: 1.79.2t1.a.a Projective image: $D_5$ Projective stem field: Galois closure of 5.1.6241.1

## Defining polynomial

 $f(x)$ $=$ $$x^{5} - x^{4} + x^{3} - 2x^{2} + 3x - 1$$ x^5 - x^4 + x^3 - 2*x^2 + 3*x - 1 .

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $$x^{2} + 6x + 3$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.