Properties

Label 2.119.5t2.a
Dimension $2$
Group $D_{5}$
Conductor $119$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{5}$
Conductor:\(119\)\(\medspace = 7 \cdot 17 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 5.1.14161.1
Galois orbit size: $2$
Smallest permutation container: $D_{5}$
Parity: odd
Projective image: $D_5$
Projective field: 5.1.14161.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \(x^{2} + 7 x + 2\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 5 a + 3 + \left(3 a + 10\right)\cdot 11 + \left(5 a + 6\right)\cdot 11^{2} + \left(9 a + 4\right)\cdot 11^{3} + \left(10 a + 8\right)\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 6 a + 1 + \left(7 a + 8\right)\cdot 11 + \left(5 a + 2\right)\cdot 11^{2} + \left(a + 4\right)\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 7 + 5\cdot 11 + 2\cdot 11^{2} + 6\cdot 11^{3} + 7\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + \left(9 a + 4\right)\cdot 11 + \left(6 a + 1\right)\cdot 11^{2} + \left(8 a + 10\right)\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 8 a + 1 + \left(a + 5\right)\cdot 11 + \left(4 a + 8\right)\cdot 11^{2} + \left(10 a + 6\right)\cdot 11^{3} + \left(2 a + 8\right)\cdot 11^{4} +O(11^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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