Properties

Label 1.76.6t1.a
Dimension $1$
Group $C_6$
Conductor $76$
Indicator $0$

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:\(76\)\(\medspace = 2^{2} \cdot 19 \)
Artin number field: Galois closure of 6.0.8340544.1
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 $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 10 a + 2 + \left(10 a + 5\right)\cdot 11 + \left(4 a + 6\right)\cdot 11^{2} + \left(3 a + 6\right)\cdot 11^{3} + \left(6 a + 5\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 a + 4 + \left(a + 6\right)\cdot 11 + \left(a + 9\right)\cdot 11^{2} + \left(5 a + 6\right)\cdot 11^{3} + \left(8 a + 7\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 7 a + 8 + \left(6 a + 6\right)\cdot 11 + 9 a\cdot 11^{2} + \left(8 a + 9\right)\cdot 11^{3} + \left(10 a + 4\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a + 9 + 5\cdot 11 + \left(6 a + 4\right)\cdot 11^{2} + \left(7 a + 4\right)\cdot 11^{3} + \left(4 a + 5\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 2 a + 7 + \left(9 a + 4\right)\cdot 11 + \left(9 a + 1\right)\cdot 11^{2} + \left(5 a + 4\right)\cdot 11^{3} + \left(2 a + 3\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 4 a + 3 + \left(4 a + 4\right)\cdot 11 + \left(a + 10\right)\cdot 11^{2} + \left(2 a + 1\right)\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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