Properties

Label 1.91.6t1.i
Dimension $1$
Group $C_6$
Conductor $91$
Indicator $0$

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:\(91\)\(\medspace = 7 \cdot 13 \)
Artin number field: Galois closure of 6.6.891474493.2
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
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:
$r_{ 1 }$ $=$ \( 7 a + 4 + \left(2 a + 3\right)\cdot 11 + 4\cdot 11^{2} + \left(8 a + 8\right)\cdot 11^{3} + 10 a\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 a + 8 + \left(a + 1\right)\cdot 11 + \left(a + 10\right)\cdot 11^{2} + \left(9 a + 10\right)\cdot 11^{3} + \left(5 a + 7\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 10 a + 10 + \left(8 a + 2\right)\cdot 11 + \left(a + 2\right)\cdot 11^{2} + \left(6 a + 7\right)\cdot 11^{3} + 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 4 a + 10 + \left(8 a + 6\right)\cdot 11 + \left(10 a + 2\right)\cdot 11^{2} + \left(2 a + 7\right)\cdot 11^{3} + 2\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a + 7 + 9 a\cdot 11 + \left(9 a + 2\right)\cdot 11^{2} + \left(a + 2\right)\cdot 11^{3} + 5 a\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( a + 6 + \left(2 a + 6\right)\cdot 11 + 9 a\cdot 11^{2} + \left(4 a + 8\right)\cdot 11^{3} + \left(10 a + 8\right)\cdot 11^{4} +O(11^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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