Properties

Label 2.324.3t2.a
Dimension $2$
Group $S_3$
Conductor $324$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$S_3$
Conductor:\(324\)\(\medspace = 2^{2} \cdot 3^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.419904.2
Galois orbit size: $1$
Smallest permutation container: $S_3$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 6.0.419904.2

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 $
Roots:
$r_{ 1 }$ $=$ $ 2 a + 2 + \left(5 a + 10\right)\cdot 11 + \left(8 a + 5\right)\cdot 11^{2} + \left(2 a + 4\right)\cdot 11^{3} + 9\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 a + 10 + \left(5 a + 6\right)\cdot 11 + \left(2 a + 1\right)\cdot 11^{2} + \left(8 a + 7\right)\cdot 11^{3} + \left(10 a + 7\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 3 a + 1 + \left(6 a + 5\right)\cdot 11 + \left(6 a + 7\right)\cdot 11^{2} + \left(4 a + 7\right)\cdot 11^{3} + \left(3 a + 5\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 6 a + 8 + \left(10 a + 6\right)\cdot 11 + \left(6 a + 8\right)\cdot 11^{2} + \left(3 a + 9\right)\cdot 11^{3} + \left(7 a + 6\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 5 a + 10 + 9\cdot 11 + \left(4 a + 3\right)\cdot 11^{2} + \left(7 a + 6\right)\cdot 11^{3} + \left(3 a + 10\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 8 a + 2 + \left(4 a + 5\right)\cdot 11 + \left(4 a + 5\right)\cdot 11^{2} + \left(6 a + 8\right)\cdot 11^{3} + \left(7 a + 3\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $2$
$3$ $2$ $(1,2)(3,6)(4,5)$ $0$
$2$ $3$ $(1,3,4)(2,5,6)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.