Properties

Label 4.137...159.10t12.a.a
Dimension $4$
Group $S_5$
Conductor $1.375\times 10^{12}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(1374665998159\)\(\medspace = 11119^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.3.11119.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: odd
Determinant: 1.11119.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.3.11119.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 2x^{4} + x^{3} + x^{2} - 3x + 1 \) Copy content Toggle raw display .

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \) Copy content Toggle raw display

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