Properties

Label 4.4169.5t5.a.a
Dimension $4$
Group $S_5$
Conductor $4169$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(4169\)\(\medspace = 11 \cdot 379 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.4169.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.4169.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.1.4169.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \( x^{2} + 6x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a + 4 + 3 a\cdot 7 + \left(a + 3\right)\cdot 7^{2} + 5\cdot 7^{3} +O(7^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 a + 5 + \left(3 a + 2\right)\cdot 7 + \left(5 a + 1\right)\cdot 7^{2} + \left(6 a + 4\right)\cdot 7^{3} + 6 a\cdot 7^{4} +O(7^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 6 + 7^{2} + 7^{3} +O(7^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + \left(5 a + 5\right)\cdot 7 + \left(2 a + 5\right)\cdot 7^{2} + \left(2 a + 1\right)\cdot 7^{3} + \left(3 a + 2\right)\cdot 7^{4} +O(7^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( a + 6 + \left(a + 4\right)\cdot 7 + \left(4 a + 2\right)\cdot 7^{2} + \left(4 a + 1\right)\cdot 7^{3} + \left(3 a + 3\right)\cdot 7^{4} +O(7^{5})\) 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 valueComplex conjugation
$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$