Properties

Label 2.131.5t2.a.b
Dimension $2$
Group $D_{5}$
Conductor $131$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{5}$
Conductor: \(131\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.1.17161.1
Galois orbit size: $2$
Smallest permutation container: $D_{5}$
Parity: odd
Determinant: 1.131.2t1.a.a
Projective image: $D_5$
Projective stem field: 5.1.17161.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

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

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$2$
$5$$2$$(1,2)(3,4)$$0$
$2$$5$$(1,4,3,2,5)$$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$
$2$$5$$(1,3,5,4,2)$$\zeta_{5}^{3} + \zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.