# Properties

 Label 4.33792250337.10t12.b.a Dimension $4$ Group $\PGL(2,5)$ Conductor $33792250337$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $4$ Group: $\PGL(2,5)$ Conductor: $$33792250337$$$$\medspace = 53^{3} \cdot 61^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 6.2.33792250337.1 Galois orbit size: $1$ Smallest permutation container: $S_5$ Parity: even Determinant: 1.3233.2t1.a.a Projective image: $S_5$ Projective stem field: 6.2.33792250337.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} - 19 x^{4} + 58 x^{3} + 220 x^{2} - 112 x - 741$$  .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{2} + 12 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$13 + 8\cdot 13^{2} + 10\cdot 13^{3} + 12\cdot 13^{4} +O(13^{5})$$ $r_{ 2 }$ $=$ $$5 a + 12 + \left(6 a + 8\right)\cdot 13 + \left(3 a + 6\right)\cdot 13^{2} + \left(12 a + 3\right)\cdot 13^{3} + \left(7 a + 4\right)\cdot 13^{4} +O(13^{5})$$ $r_{ 3 }$ $=$ $$4 + 2\cdot 13 + 8\cdot 13^{2} + 12\cdot 13^{3} + 9\cdot 13^{4} +O(13^{5})$$ $r_{ 4 }$ $=$ $$8 a + 6 + \left(3 a + 10\right)\cdot 13 + \left(7 a + 10\right)\cdot 13^{2} + \left(2 a + 8\right)\cdot 13^{3} + \left(4 a + 11\right)\cdot 13^{4} +O(13^{5})$$ $r_{ 5 }$ $=$ $$8 a + 4 + \left(6 a + 10\right)\cdot 13 + \left(9 a + 3\right)\cdot 13^{2} + 12\cdot 13^{3} + \left(5 a + 12\right)\cdot 13^{4} +O(13^{5})$$ $r_{ 6 }$ $=$ $$5 a + 1 + \left(9 a + 6\right)\cdot 13 + \left(5 a + 1\right)\cdot 13^{2} + \left(10 a + 4\right)\cdot 13^{3} + 8 a\cdot 13^{4} +O(13^{5})$$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $10$ $2$ $(1,3)(2,4)(5,6)$ $-2$ $15$ $2$ $(1,3)(4,6)$ $0$ $20$ $3$ $(1,6,5)(2,3,4)$ $1$ $30$ $4$ $(1,4,3,6)$ $0$ $24$ $5$ $(1,5,4,6,2)$ $-1$ $20$ $6$ $(1,4,6,2,5,3)$ $1$

The blue line marks the conjugacy class containing complex conjugation.