Properties

Label 1.19.6t1.a
Dimension $1$
Group $C_6$
Conductor $19$
Indicator $0$

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:\(19\)
Artin number field: Galois closure of 6.0.2476099.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 17 a + 22 + \left(15 a + 9\right)\cdot 31 + \left(27 a + 14\right)\cdot 31^{2} + \left(15 a + 29\right)\cdot 31^{3} + \left(23 a + 19\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 14 a + 25 + \left(15 a + 23\right)\cdot 31 + \left(3 a + 22\right)\cdot 31^{2} + \left(15 a + 2\right)\cdot 31^{3} + \left(7 a + 20\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 10 a + 23 + \left(27 a + 13\right)\cdot 31 + \left(8 a + 26\right)\cdot 31^{2} + \left(15 a + 4\right)\cdot 31^{3} + \left(18 a + 30\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 21 a + 12 + \left(3 a + 27\right)\cdot 31 + \left(22 a + 16\right)\cdot 31^{2} + \left(15 a + 26\right)\cdot 31^{3} + \left(12 a + 20\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 11 a + 26 + 29\cdot 31 + \left(23 a + 29\right)\cdot 31^{2} + \left(17 a + 23\right)\cdot 31^{3} + \left(30 a + 25\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 20 a + 17 + \left(30 a + 19\right)\cdot 31 + \left(7 a + 13\right)\cdot 31^{2} + \left(13 a + 5\right)\cdot 31^{3} + 7\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,6,4,2,5,3)$
$(1,2)(3,4)(5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $1$ $1$
$1$ $2$ $(1,2)(3,4)(5,6)$ $-1$ $-1$
$1$ $3$ $(1,4,5)(2,3,6)$ $\zeta_{3}$ $-\zeta_{3} - 1$
$1$ $3$ $(1,5,4)(2,6,3)$ $-\zeta_{3} - 1$ $\zeta_{3}$
$1$ $6$ $(1,6,4,2,5,3)$ $\zeta_{3} + 1$ $-\zeta_{3}$
$1$ $6$ $(1,3,5,2,4,6)$ $-\zeta_{3}$ $\zeta_{3} + 1$
The blue line marks the conjugacy class containing complex conjugation.