Properties

Label 5.101...853.6t14.a.a
Dimension $5$
Group $S_5$
Conductor $1.017\times 10^{16}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(10167118812786853\)\(\medspace = 23^{3} \cdot 9419^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.216637.1
Galois orbit size: $1$
Smallest permutation container: $\PGL(2,5)$
Parity: even
Determinant: 1.216637.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.216637.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 2x^{4} - 4x^{3} + 7x^{2} + x - 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 4 a + 5 + 3\cdot 11 + \left(10 a + 7\right)\cdot 11^{2} + \left(8 a + 10\right)\cdot 11^{3} + \left(8 a + 7\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 10 + 10 a\cdot 11 + 3\cdot 11^{2} + \left(2 a + 3\right)\cdot 11^{3} + \left(2 a + 1\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( a + 5 + 5\cdot 11 + \left(3 a + 8\right)\cdot 11^{2} + \left(6 a + 6\right)\cdot 11^{3} + \left(10 a + 8\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 + 7\cdot 11 + 4\cdot 11^{2} + 5\cdot 11^{3} + 3\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 10 a + 9 + \left(10 a + 4\right)\cdot 11 + \left(7 a + 9\right)\cdot 11^{2} + \left(4 a + 6\right)\cdot 11^{3} +O(11^{5})\) 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$$1$$()$$5$
$10$$2$$(1,2)$$-1$
$15$$2$$(1,2)(3,4)$$1$
$20$$3$$(1,2,3)$$-1$
$30$$4$$(1,2,3,4)$$1$
$24$$5$$(1,2,3,4,5)$$0$
$20$$6$$(1,2,3)(4,5)$$-1$

The blue line marks the conjugacy class containing complex conjugation.