# Properties

 Label 3.105625.12t33.a.a Dimension $3$ Group $A_5$ Conductor $105625$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $$105625$$$$\medspace = 5^{4} \cdot 13^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 5.1.2640625.1 Galois orbit size: $2$ Smallest permutation container: $A_5$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_5$ Projective field: Galois closure of 5.1.2640625.1

## Defining polynomial

 $f(x)$ $=$ $x^{5} + 5 x^{3} - 10 x^{2} - 45$.

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $x^{2} + 33 x + 2$

Roots:
 $r_{ 1 }$ $=$ $16 + 34\cdot 37^{2} + 11\cdot 37^{3} + 15\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 2 }$ $=$ $23 a + 29 + \left(4 a + 30\right)\cdot 37 + \left(6 a + 24\right)\cdot 37^{2} + \left(7 a + 3\right)\cdot 37^{3} + \left(32 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 3 }$ $=$ $14 a + 10 + \left(32 a + 26\right)\cdot 37 + \left(30 a + 7\right)\cdot 37^{2} + \left(29 a + 26\right)\cdot 37^{3} + \left(4 a + 32\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 4 }$ $=$ $14 a + \left(9 a + 15\right)\cdot 37 + \left(35 a + 30\right)\cdot 37^{2} + \left(12 a + 7\right)\cdot 37^{3} + \left(3 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 5 }$ $=$ $23 a + 19 + \left(27 a + 1\right)\cdot 37 + \left(a + 14\right)\cdot 37^{2} + \left(24 a + 24\right)\cdot 37^{3} + \left(33 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character value $1$ $1$ $()$ $3$ $15$ $2$ $(1,2)(3,4)$ $-1$ $20$ $3$ $(1,2,3)$ $0$ $12$ $5$ $(1,2,3,4,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $12$ $5$ $(1,3,4,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$

The blue line marks the conjugacy class containing complex conjugation.