Properties

Label 8.35101146609.12t213.a.a
Dimension $8$
Group $S_3\wr S_3$
Conductor $35101146609$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $S_3\wr S_3$
Conductor: \(35101146609\)\(\medspace = 3^{12} \cdot 257^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.7.334110914019.1
Galois orbit size: $1$
Smallest permutation container: 12T213
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_3\wr S_3$
Projective stem field: Galois closure of 9.7.334110914019.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 9x^{7} - x^{6} + 27x^{5} + 6x^{4} - 31x^{3} - 9x^{2} + 12x + 3 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: \( x^{3} + 9x + 76 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 63 a^{2} + 78 a + 54 + \left(24 a^{2} + 63 a + 67\right)\cdot 79 + \left(31 a^{2} + 51 a + 13\right)\cdot 79^{2} + \left(18 a^{2} + 19 a + 43\right)\cdot 79^{3} + \left(60 a^{2} + 67 a + 24\right)\cdot 79^{4} + \left(28 a^{2} + 60 a + 29\right)\cdot 79^{5} + \left(22 a^{2} + a + 75\right)\cdot 79^{6} + \left(a^{2} + 56 a + 11\right)\cdot 79^{7} + \left(61 a^{2} + 3 a + 77\right)\cdot 79^{8} + \left(10 a^{2} + 7 a + 25\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 47 a^{2} + 61 a + 67 + \left(78 a^{2} + 30 a + 21\right)\cdot 79 + \left(18 a^{2} + 69 a + 52\right)\cdot 79^{2} + \left(54 a^{2} + 57 a + 59\right)\cdot 79^{3} + \left(31 a^{2} + 11 a + 5\right)\cdot 79^{4} + \left(66 a^{2} + 76 a + 21\right)\cdot 79^{5} + \left(16 a^{2} + 15 a + 63\right)\cdot 79^{6} + \left(31 a^{2} + 28 a + 74\right)\cdot 79^{7} + \left(59 a^{2} + 62 a + 62\right)\cdot 79^{8} + \left(22 a^{2} + 42 a + 49\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 39 a^{2} + 34 a + 68 + \left(49 a^{2} + 11 a + 57\right)\cdot 79 + \left(11 a^{2} + 10 a + 53\right)\cdot 79^{2} + \left(76 a^{2} + 37 a + 73\right)\cdot 79^{3} + \left(36 a^{2} + 34 a + 42\right)\cdot 79^{4} + \left(22 a^{2} + 5 a + 70\right)\cdot 79^{5} + \left(51 a^{2} + 50 a + 11\right)\cdot 79^{6} + \left(8 a^{2} + 23 a + 56\right)\cdot 79^{7} + \left(71 a^{2} + 42 a + 58\right)\cdot 79^{8} + \left(54 a^{2} + 72 a + 53\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 55 a^{2} + 51 a + \left(74 a^{2} + 44 a + 31\right)\cdot 79 + \left(23 a^{2} + 71 a + 63\right)\cdot 79^{2} + \left(40 a^{2} + 77 a + 21\right)\cdot 79^{3} + \left(65 a^{2} + 10 a + 45\right)\cdot 79^{4} + \left(63 a^{2} + 69 a + 34\right)\cdot 79^{5} + \left(56 a^{2} + 35 a + 42\right)\cdot 79^{6} + \left(57 a^{2} + 51 a + 59\right)\cdot 79^{7} + \left(72 a^{2} + 62 a + 70\right)\cdot 79^{8} + \left(42 a^{2} + 36 a + 66\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 41 a^{2} + 18 a + 31 + \left(36 a^{2} + 57 a + 6\right)\cdot 79 + \left(25 a^{2} + 44 a + 12\right)\cdot 79^{2} + \left(33 a^{2} + 20 a + 13\right)\cdot 79^{3} + \left(49 a^{2} + 7 a + 33\right)\cdot 79^{4} + \left(2 a^{2} + 4 a + 33\right)\cdot 79^{5} + \left(61 a^{2} + 49 a + 12\right)\cdot 79^{6} + \left(41 a^{2} + 17 a + 59\right)\cdot 79^{7} + \left(42 a^{2} + 35 a + 40\right)\cdot 79^{8} + \left(39 a^{2} + 40 a + 71\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 49 a^{2} + 45 a + 43 + \left(65 a^{2} + 76 a + 55\right)\cdot 79 + \left(32 a^{2} + 24 a + 37\right)\cdot 79^{2} + \left(11 a^{2} + 41 a + 6\right)\cdot 79^{3} + \left(44 a^{2} + 63 a + 75\right)\cdot 79^{4} + \left(46 a^{2} + 74 a + 9\right)\cdot 79^{5} + \left(26 a^{2} + 14 a + 19\right)\cdot 79^{6} + \left(64 a^{2} + 22 a + 20\right)\cdot 79^{7} + \left(30 a^{2} + 55 a + 56\right)\cdot 79^{8} + \left(7 a^{2} + 10 a + 11\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 56 a^{2} + 46 a + 12 + \left(4 a^{2} + 3 a + 26\right)\cdot 79 + \left(36 a^{2} + 17 a + 42\right)\cdot 79^{2} + \left(63 a^{2} + 22 a + 76\right)\cdot 79^{3} + \left(60 a^{2} + 56 a + 27\right)\cdot 79^{4} + \left(27 a^{2} + 12 a + 23\right)\cdot 79^{5} + \left(5 a^{2} + 27 a + 52\right)\cdot 79^{6} + \left(69 a^{2} + 78 a + 23\right)\cdot 79^{7} + \left(25 a^{2} + 32 a + 24\right)\cdot 79^{8} + \left(13 a^{2} + 78 a + 41\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 54 a^{2} + 62 a + 73 + \left(17 a^{2} + 36 a + 4\right)\cdot 79 + \left(22 a^{2} + 61 a + 53\right)\cdot 79^{2} + \left(27 a^{2} + 38 a + 22\right)\cdot 79^{3} + \left(48 a^{2} + 4 a + 21\right)\cdot 79^{4} + \left(47 a^{2} + 14 a + 16\right)\cdot 79^{5} + \left(74 a^{2} + 28 a + 70\right)\cdot 79^{6} + \left(35 a^{2} + 5 a + 7\right)\cdot 79^{7} + \left(54 a^{2} + 40 a + 40\right)\cdot 79^{8} + \left(28 a^{2} + 31 a + 60\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 70 a^{2} + 47 + \left(42 a^{2} + 70 a + 44\right)\cdot 79 + \left(34 a^{2} + 43 a + 66\right)\cdot 79^{2} + \left(70 a^{2} + 77\right)\cdot 79^{3} + \left(76 a^{2} + 60 a + 39\right)\cdot 79^{4} + \left(9 a^{2} + 77 a + 77\right)\cdot 79^{5} + \left(a^{2} + 13 a + 47\right)\cdot 79^{6} + \left(6 a^{2} + 33 a + 2\right)\cdot 79^{7} + \left(56 a^{2} + 60 a + 43\right)\cdot 79^{8} + \left(16 a^{2} + 74 a + 13\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

Cycle notation
$(1,5,8)$
$(2,4)$
$(1,2)(4,5)(7,8)$
$(1,3,2)(4,5,6)(7,8,9)$
$(2,4,7)$
$(3,6,9)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$8$
$9$$2$$(2,4)$$0$
$18$$2$$(1,2)(4,5)(7,8)$$4$
$27$$2$$(1,5)(2,4)(3,6)$$0$
$27$$2$$(2,4)(3,6)$$0$
$54$$2$$(1,3)(2,4)(5,6)(8,9)$$0$
$6$$3$$(3,6,9)$$-4$
$8$$3$$(1,8,5)(2,7,4)(3,9,6)$$-1$
$12$$3$$(1,5,8)(3,6,9)$$2$
$72$$3$$(1,3,2)(4,5,6)(7,8,9)$$2$
$54$$4$$(2,3,4,6)(7,9)$$0$
$162$$4$$(2,3,4,6)(5,8)(7,9)$$0$
$36$$6$$(1,2)(3,6,9)(4,5)(7,8)$$-2$
$36$$6$$(2,3,7,9,4,6)$$-2$
$36$$6$$(2,4)(3,6,9)$$0$
$36$$6$$(1,5,8)(2,4)(3,6,9)$$0$
$54$$6$$(1,8,5)(2,4)(3,6)$$0$
$72$$6$$(1,4,5,7,8,2)(3,6,9)$$1$
$108$$6$$(1,3,5,6,8,9)(2,4)$$0$
$216$$6$$(1,3,4,5,6,2)(7,8,9)$$0$
$144$$9$$(1,3,7,8,9,4,5,6,2)$$-1$
$108$$12$$(1,5,8)(2,3,4,6)(7,9)$$0$

The blue line marks the conjugacy class containing complex conjugation.