Properties

Label 1.152.6t1.c
Dimension $1$
Group $C_6$
Conductor $152$
Indicator $0$

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:\(152\)\(\medspace = 2^{3} \cdot 19 \)
Artin number field: Galois closure of 6.0.66724352.1
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 $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \( x^{2} + 6x + 3 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 3 a + 6 a\cdot 7 + 4 a\cdot 7^{2} + \left(6 a + 5\right)\cdot 7^{3} + \left(a + 1\right)\cdot 7^{4} +O(7^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 a + 5 + \left(6 a + 4\right)\cdot 7 + \left(4 a + 1\right)\cdot 7^{2} + \left(6 a + 6\right)\cdot 7^{3} + \left(a + 3\right)\cdot 7^{4} +O(7^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 4 a + 3 + 3\cdot 7 + \left(2 a + 5\right)\cdot 7^{2} + 6\cdot 7^{3} + \left(5 a + 3\right)\cdot 7^{4} +O(7^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 4 a + 5 + \left(2 a + 6\right)\cdot 7^{2} + 7^{3} + 5 a\cdot 7^{4} +O(7^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 4 a + 1 + 7 + 2 a\cdot 7^{2} + 7^{3} + \left(5 a + 6\right)\cdot 7^{4} +O(7^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 3 a + 2 + \left(6 a + 4\right)\cdot 7 + 4 a\cdot 7^{2} + 6 a\cdot 7^{3} + \left(a + 5\right)\cdot 7^{4} +O(7^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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