Properties

Label 6.32941720000.42t82.a.a
Dimension $6$
Group $\PGL(2,7)$
Conductor $32941720000$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $\PGL(2,7)$
Conductor: \(32941720000\)\(\medspace = 2^{6} \cdot 5^{4} \cdot 7^{7} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.32941720000.1
Galois orbit size: $1$
Smallest permutation container: 42T82
Parity: odd
Determinant: 1.7.2t1.a.a
Projective image: $\PGL(2,7)$
Projective stem field: Galois closure of 8.2.32941720000.1

Defining polynomial

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

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$6$
$21$$2$$(1,6)(2,7)(3,4)(5,8)$$-2$
$28$$2$$(2,5)(3,8)(6,7)$$0$
$56$$3$$(2,3,7)(5,8,6)$$0$
$42$$4$$(1,4,8,5)(2,6,3,7)$$2$
$56$$6$$(2,6,3,5,7,8)$$0$
$48$$7$$(1,6,3,5,4,7,2)$$-1$
$42$$8$$(1,6,4,3,8,7,5,2)$$0$
$42$$8$$(1,3,5,6,8,2,4,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.