Properties

Label 2.143.5t2.a
Dimension $2$
Group $D_{5}$
Conductor $143$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{5}$
Conductor:\(143\)\(\medspace = 11 \cdot 13 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 5.1.20449.1
Galois orbit size: $2$
Smallest permutation container: $D_{5}$
Parity: odd
Projective image: $D_5$
Projective field: 5.1.20449.1

Galois action

Roots of defining polynomial

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:
$r_{ 1 }$ $=$ \( 13 a + 3 + \left(2 a + 11\right)\cdot 17 + \left(13 a + 12\right)\cdot 17^{2} + \left(8 a + 9\right)\cdot 17^{3} + \left(13 a + 16\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 9 a + 7 + \left(a + 8\right)\cdot 17 + \left(8 a + 5\right)\cdot 17^{2} + \left(a + 16\right)\cdot 17^{3} + \left(11 a + 2\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 10 + 12\cdot 17 + 14\cdot 17^{2} + 9\cdot 17^{3} + 14\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 8 a + 16 + 15 a\cdot 17 + \left(8 a + 12\right)\cdot 17^{2} + \left(15 a + 9\right)\cdot 17^{3} + \left(5 a + 12\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 4 a + 16 + 14 a\cdot 17 + \left(3 a + 6\right)\cdot 17^{2} + \left(8 a + 5\right)\cdot 17^{3} + \left(3 a + 4\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$5$ $2$ $(1,3)(2,5)$ $0$ $0$
$2$ $5$ $(1,5,2,3,4)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$
$2$ $5$ $(1,2,4,5,3)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $\zeta_{5}^{3} + \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.