Properties

Label 3.29e3_229.6t11.2c1
Dimension 3
Group $S_4\times C_2$
Conductor $ 29^{3} \cdot 229 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$S_4\times C_2$
Conductor:$5585081= 29^{3} \cdot 229 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - x^{4} - 18 x^{3} - 211 x^{2} - 415 x - 538 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4\times C_2$
Parity: Even
Determinant: 1.29_229.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 11 + 37\cdot 53 + 49\cdot 53^{2} + 32\cdot 53^{3} + 39\cdot 53^{4} + 6\cdot 53^{5} + 34\cdot 53^{6} + 7\cdot 53^{7} + 7\cdot 53^{8} + 12\cdot 53^{9} + 23\cdot 53^{10} + 17\cdot 53^{11} +O\left(53^{ 12 }\right)$
$r_{ 2 }$ $=$ $ 11 a + 16 + \left(41 a + 52\right)\cdot 53 + \left(7 a + 18\right)\cdot 53^{2} + \left(38 a + 11\right)\cdot 53^{3} + \left(16 a + 30\right)\cdot 53^{4} + \left(26 a + 34\right)\cdot 53^{5} + \left(30 a + 34\right)\cdot 53^{6} + \left(7 a + 17\right)\cdot 53^{7} + \left(13 a + 16\right)\cdot 53^{8} + \left(4 a + 18\right)\cdot 53^{9} + \left(24 a + 15\right)\cdot 53^{10} + \left(26 a + 30\right)\cdot 53^{11} +O\left(53^{ 12 }\right)$
$r_{ 3 }$ $=$ $ 42 a + 7 + \left(11 a + 47\right)\cdot 53 + \left(45 a + 8\right)\cdot 53^{2} + \left(14 a + 50\right)\cdot 53^{3} + \left(36 a + 5\right)\cdot 53^{4} + \left(26 a + 17\right)\cdot 53^{5} + \left(22 a + 24\right)\cdot 53^{6} + \left(45 a + 17\right)\cdot 53^{7} + \left(39 a + 8\right)\cdot 53^{8} + \left(48 a + 22\right)\cdot 53^{9} + \left(28 a + 1\right)\cdot 53^{10} + \left(26 a + 6\right)\cdot 53^{11} +O\left(53^{ 12 }\right)$
$r_{ 4 }$ $=$ $ 33 + 6\cdot 53 + 20\cdot 53^{2} + 28\cdot 53^{3} + 22\cdot 53^{4} + 45\cdot 53^{5} + 35\cdot 53^{6} + 5\cdot 53^{7} + 46\cdot 53^{8} + 44\cdot 53^{9} + 40\cdot 53^{10} + 7\cdot 53^{11} +O\left(53^{ 12 }\right)$
$r_{ 5 }$ $=$ $ 2 a + 43 + \left(33 a + 48\right)\cdot 53 + \left(50 a + 51\right)\cdot 53^{2} + \left(44 a + 32\right)\cdot 53^{3} + \left(24 a + 29\right)\cdot 53^{4} + \left(50 a + 18\right)\cdot 53^{5} + \left(50 a + 44\right)\cdot 53^{6} + \left(31 a + 16\right)\cdot 53^{7} + \left(5 a + 45\right)\cdot 53^{8} + \left(24 a + 11\right)\cdot 53^{9} + \left(29 a + 45\right)\cdot 53^{10} + \left(12 a + 11\right)\cdot 53^{11} +O\left(53^{ 12 }\right)$
$r_{ 6 }$ $=$ $ 51 a + 51 + \left(19 a + 19\right)\cdot 53 + \left(2 a + 9\right)\cdot 53^{2} + \left(8 a + 3\right)\cdot 53^{3} + \left(28 a + 31\right)\cdot 53^{4} + \left(2 a + 36\right)\cdot 53^{5} + \left(2 a + 38\right)\cdot 53^{6} + \left(21 a + 40\right)\cdot 53^{7} + \left(47 a + 35\right)\cdot 53^{8} + \left(28 a + 49\right)\cdot 53^{9} + \left(23 a + 32\right)\cdot 53^{10} + \left(40 a + 32\right)\cdot 53^{11} +O\left(53^{ 12 }\right)$

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

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

Character values on conjugacy classes

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