# Properties

 Label 2.1444.5t2.a.a Dimension $2$ Group $D_{5}$ Conductor $1444$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{5}$ Conductor: $$1444$$$$\medspace = 2^{2} \cdot 19^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 5.1.2085136.1 Galois orbit size: $2$ Smallest permutation container: $D_{5}$ Parity: odd Determinant: 1.4.2t1.a.a Projective image: $D_5$ Projective stem field: 5.1.2085136.1

## Defining polynomial

 $f(x)$ $=$ $$x^{5} - 2 x^{4} - 6 x^{3} + 10 x^{2} + 17 x - 12$$  .

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

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

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

## 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 $1$ $1$ $()$ $2$ $5$ $2$ $(1,3)(2,5)$ $0$ $2$ $5$ $(1,3,2,4,5)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $2$ $5$ $(1,2,5,3,4)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$

The blue line marks the conjugacy class containing complex conjugation.