Properties

Label 3.53824.12t33.b.b
Dimension $3$
Group $A_5$
Conductor $53824$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_5$
Conductor: \(53824\)\(\medspace = 2^{6} \cdot 29^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.45265984.1
Galois orbit size: $2$
Smallest permutation container: $A_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_5$
Projective stem field: Galois closure of 5.1.45265984.1

Defining polynomial

$f(x)$$=$ \( x^{5} - x^{4} + 12x^{3} - 14x^{2} + 42x - 62 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 a + 3 + \left(14 a + 6\right)\cdot 17 + \left(12 a + 10\right)\cdot 17^{2} + \left(8 a + 8\right)\cdot 17^{3} + \left(7 a + 6\right)\cdot 17^{4} + \left(12 a + 5\right)\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 + 14\cdot 17 + 7\cdot 17^{2} + 15\cdot 17^{3} + 10\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 a + 11 + \left(2 a + 12\right)\cdot 17 + \left(4 a + 8\right)\cdot 17^{2} + \left(8 a + 4\right)\cdot 17^{3} + \left(9 a + 5\right)\cdot 17^{4} + \left(4 a + 10\right)\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 13 + \left(a + 9\right)\cdot 17 + 7 a\cdot 17^{2} + \left(a + 14\right)\cdot 17^{3} + \left(10 a + 14\right)\cdot 17^{4} + \left(12 a + 2\right)\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 14 a + 16 + \left(15 a + 7\right)\cdot 17 + \left(9 a + 6\right)\cdot 17^{2} + \left(15 a + 8\right)\cdot 17^{3} + \left(6 a + 6\right)\cdot 17^{4} + \left(4 a + 5\right)\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display

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

SizeOrderAction 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} + 1$
$12$$5$$(1,3,4,5,2)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.