Properties

Label 3.17101.6t6.a.a
Dimension 3
Group $A_4\times C_2$
Conductor $ 7^{2} \cdot 349 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_4\times C_2$
Conductor:$17101= 7^{2} \cdot 349 $
Artin number field: Splitting field of 6.2.837949.1 defined by $f= x^{6} - 2 x^{5} + 3 x^{4} - 9 x^{3} + 4 x^{2} - 9 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_4\times C_2$
Parity: Even
Determinant: 1.349.2t1.a.a
Projective image: $A_4$
Projective field: Galois closure of 4.0.5968249.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 24 a + \left(13 a + 4\right)\cdot 41 + \left(24 a + 26\right)\cdot 41^{2} + \left(34 a + 2\right)\cdot 41^{3} + \left(28 a + 36\right)\cdot 41^{4} + \left(30 a + 15\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 17 a + 31 + \left(27 a + 20\right)\cdot 41 + \left(16 a + 3\right)\cdot 41^{2} + 6 a\cdot 41^{3} + \left(12 a + 6\right)\cdot 41^{4} + \left(10 a + 38\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 18 + 31\cdot 41 + 28\cdot 41^{2} + 18\cdot 41^{3} + 34\cdot 41^{4} + 38\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 11 a + 23 + \left(11 a + 22\right)\cdot 41 + \left(12 a + 35\right)\cdot 41^{2} + \left(28 a + 30\right)\cdot 41^{3} + \left(3 a + 13\right)\cdot 41^{4} + \left(4 a + 29\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 38 + 39\cdot 41 + 8\cdot 41^{2} + 8\cdot 41^{3} + 36\cdot 41^{4} + 3\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 30 a + 15 + \left(29 a + 4\right)\cdot 41 + \left(28 a + 20\right)\cdot 41^{2} + \left(12 a + 21\right)\cdot 41^{3} + \left(37 a + 37\right)\cdot 41^{4} + \left(36 a + 37\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$

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

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

Character values on conjugacy classes

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