Properties

Label 4.12447.5t5.a.a
Dimension $4$
Group $S_5$
Conductor $12447$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(12447\)\(\medspace = 3^{3} \cdot 461 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.3.12447.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: odd
Determinant: 1.1383.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.3.12447.1

Defining polynomial

$f(x)$$=$ \( x^{5} + x^{3} - x^{2} - 3x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 17 + 30\cdot 47 + 32\cdot 47^{2} + 11\cdot 47^{3} + 39\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 38 a + 22 + \left(24 a + 32\right)\cdot 47 + \left(35 a + 17\right)\cdot 47^{2} + \left(29 a + 35\right)\cdot 47^{3} + \left(3 a + 6\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 a + 4 + \left(22 a + 44\right)\cdot 47 + \left(11 a + 16\right)\cdot 47^{2} + \left(17 a + 12\right)\cdot 47^{3} + \left(43 a + 31\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 13 a + 36 + \left(7 a + 39\right)\cdot 47 + \left(17 a + 46\right)\cdot 47^{2} + \left(43 a + 5\right)\cdot 47^{3} + \left(13 a + 16\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 34 a + 15 + \left(39 a + 41\right)\cdot 47 + \left(29 a + 26\right)\cdot 47^{2} + \left(3 a + 28\right)\cdot 47^{3} + 33 a\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.