Properties

Label 3.17712697.6t11.b.a
Dimension 3
Group $S_4\times C_2$
Conductor $ 41^{3} \cdot 257 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$S_4\times C_2$
Conductor:$17712697= 41^{3} \cdot 257 $
Artin number field: Splitting field of 6.2.4552163129.1 defined by $f= x^{6} - 2 x^{5} + 18 x^{4} + 56 x^{3} - 257 x^{2} - 794 x - 1671 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4\times C_2$
Parity: Even
Determinant: 1.10537.2t1.a.a
Projective image: $S_4$
Projective field: Galois closure of 4.0.257.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 4 + 52\cdot 61 + 22\cdot 61^{2} + 14\cdot 61^{3} + 56\cdot 61^{4} + 52\cdot 61^{5} + 34\cdot 61^{6} + 35\cdot 61^{7} + 55\cdot 61^{8} + 33\cdot 61^{10} + 59\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 2 }$ $=$ $ 49 a + 51 + \left(37 a + 58\right)\cdot 61 + \left(20 a + 51\right)\cdot 61^{2} + 46 a\cdot 61^{3} + \left(54 a + 21\right)\cdot 61^{4} + \left(6 a + 56\right)\cdot 61^{5} + \left(5 a + 36\right)\cdot 61^{6} + \left(17 a + 30\right)\cdot 61^{7} + \left(44 a + 17\right)\cdot 61^{8} + \left(41 a + 24\right)\cdot 61^{9} + \left(48 a + 27\right)\cdot 61^{10} + \left(51 a + 59\right)\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 3 }$ $=$ $ 59 a + 9 + \left(48 a + 32\right)\cdot 61 + \left(16 a + 59\right)\cdot 61^{2} + \left(36 a + 41\right)\cdot 61^{3} + \left(5 a + 39\right)\cdot 61^{4} + \left(57 a + 8\right)\cdot 61^{5} + \left(25 a + 4\right)\cdot 61^{6} + \left(4 a + 22\right)\cdot 61^{7} + \left(34 a + 53\right)\cdot 61^{8} + \left(3 a + 49\right)\cdot 61^{9} + \left(44 a + 8\right)\cdot 61^{10} + \left(24 a + 33\right)\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 4 }$ $=$ $ 14 + 8\cdot 61 + 29\cdot 61^{2} + 39\cdot 61^{3} + 27\cdot 61^{4} + 57\cdot 61^{5} + 37\cdot 61^{6} + 51\cdot 61^{7} + 50\cdot 61^{8} + 5\cdot 61^{9} + 30\cdot 61^{10} + 15\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 5 }$ $=$ $ 2 a + 7 + \left(12 a + 29\right)\cdot 61 + \left(4 a + 2\right)\cdot 61^{2} + \left(24 a + 34\right)\cdot 61^{3} + \left(55 a + 8\right)\cdot 61^{4} + \left(3 a + 60\right)\cdot 61^{5} + \left(35 a + 33\right)\cdot 61^{6} + 56 a\cdot 61^{7} + \left(26 a + 22\right)\cdot 61^{8} + \left(57 a + 19\right)\cdot 61^{9} + \left(16 a + 49\right)\cdot 61^{10} + \left(36 a + 13\right)\cdot 61^{11} +O\left(61^{ 12 }\right)$
$r_{ 6 }$ $=$ $ 12 a + 39 + \left(58 a + 28\right)\cdot 61 + \left(3 a + 16\right)\cdot 61^{2} + \left(15 a + 54\right)\cdot 61^{3} + \left(6 a + 29\right)\cdot 61^{4} + \left(54 a + 8\right)\cdot 61^{5} + \left(55 a + 35\right)\cdot 61^{6} + \left(43 a + 42\right)\cdot 61^{7} + \left(16 a + 44\right)\cdot 61^{8} + \left(19 a + 21\right)\cdot 61^{9} + \left(12 a + 34\right)\cdot 61^{10} + \left(9 a + 1\right)\cdot 61^{11} +O\left(61^{ 12 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$3$
$1$$2$$(1,4)(2,6)(3,5)$$-3$
$3$$2$$(3,5)$$1$
$3$$2$$(2,6)(3,5)$$-1$
$6$$2$$(1,2)(4,6)$$1$
$6$$2$$(1,2)(3,5)(4,6)$$-1$
$8$$3$$(1,3,2)(4,5,6)$$0$
$6$$4$$(2,3,6,5)$$1$
$6$$4$$(1,4)(2,3,6,5)$$-1$
$8$$6$$(1,3,6,4,5,2)$$0$
The blue line marks the conjugacy class containing complex conjugation.