Properties

Label 7.205885750000.8t43.a.a
Dimension $7$
Group $\PGL(2,7)$
Conductor $205885750000$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $7$
Group: $\PGL(2,7)$
Conductor: \(205885750000\)\(\medspace = 2^{4} \cdot 5^{6} \cdot 7^{7} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.205885750000.1
Galois orbit size: $1$
Smallest permutation container: $\PGL(2,7)$
Parity: odd
Determinant: 1.7.2t1.a.a
Projective image: $\PGL(2,7)$
Projective stem field: Galois closure of 8.2.205885750000.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 35x^{4} - 50x + 25 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{3} + 2x + 27 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 12 a^{2} + 18 a + 17 + \left(11 a^{2} + 21 a + 23\right)\cdot 29 + \left(4 a^{2} + 22 a + 13\right)\cdot 29^{2} + \left(7 a^{2} + 13 a + 10\right)\cdot 29^{3} + \left(28 a^{2} + 21 a + 1\right)\cdot 29^{4} + \left(20 a^{2} + 27 a + 3\right)\cdot 29^{5} + \left(a^{2} + 20 a + 28\right)\cdot 29^{6} + \left(14 a^{2} + 25 a + 20\right)\cdot 29^{7} + \left(27 a^{2} + 6 a + 13\right)\cdot 29^{8} + \left(9 a^{2} + 14 a + 10\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 a^{2} + 22 a + \left(5 a^{2} + 11 a + 20\right)\cdot 29 + \left(11 a^{2} + 15 a + 15\right)\cdot 29^{2} + \left(9 a^{2} + 21 a + 10\right)\cdot 29^{3} + \left(5 a^{2} + 26 a + 6\right)\cdot 29^{4} + \left(7 a^{2} + 14 a + 15\right)\cdot 29^{5} + \left(15 a^{2} + 11\right)\cdot 29^{6} + \left(9 a^{2} + a + 9\right)\cdot 29^{7} + \left(2 a^{2} + 19 a + 26\right)\cdot 29^{8} + \left(28 a^{2} + 18 a\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 14 + 10\cdot 29 + 7\cdot 29^{2} + 16\cdot 29^{4} + 29^{5} + 3\cdot 29^{6} + 13\cdot 29^{7} + 15\cdot 29^{8} + 10\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 8 a^{2} + 2 a + 18 + \left(4 a^{2} + 2 a + 18\right)\cdot 29 + \left(11 a^{2} + 10 a + 15\right)\cdot 29^{2} + \left(21 a^{2} + 26\right)\cdot 29^{3} + \left(23 a^{2} + 23 a + 1\right)\cdot 29^{4} + \left(20 a^{2} + 7 a + 14\right)\cdot 29^{5} + \left(19 a^{2} + 3 a + 17\right)\cdot 29^{6} + \left(12 a^{2} + 26 a + 13\right)\cdot 29^{7} + \left(14 a^{2} + 26 a + 13\right)\cdot 29^{8} + \left(28 a^{2} + 1\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 23 a^{2} + 18 a + 22 + \left(18 a^{2} + 21 a + 23\right)\cdot 29 + \left(25 a^{2} + 5 a + 22\right)\cdot 29^{2} + \left(23 a^{2} + 22 a + 3\right)\cdot 29^{3} + \left(24 a^{2} + 4 a + 16\right)\cdot 29^{4} + \left(13 a^{2} + 12 a + 22\right)\cdot 29^{5} + \left(27 a^{2} + 5 a + 23\right)\cdot 29^{6} + \left(3 a^{2} + 27 a + 26\right)\cdot 29^{7} + \left(9 a^{2} + 9 a + 27\right)\cdot 29^{8} + \left(12 a^{2} + 21 a + 3\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 23 a^{2} + 22 a + 22 + \left(27 a^{2} + 14 a + 6\right)\cdot 29 + \left(27 a^{2} + 16\right)\cdot 29^{2} + \left(26 a^{2} + 22 a + 17\right)\cdot 29^{3} + \left(4 a^{2} + 2 a + 18\right)\cdot 29^{4} + \left(23 a^{2} + 18 a + 15\right)\cdot 29^{5} + \left(28 a^{2} + 2 a + 25\right)\cdot 29^{6} + \left(10 a^{2} + 5 a + 16\right)\cdot 29^{7} + \left(21 a^{2} + 12 a + 5\right)\cdot 29^{8} + \left(6 a^{2} + 22 a + 6\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 23 + 12\cdot 29 + 24\cdot 29^{2} + 2\cdot 29^{3} + 8\cdot 29^{4} + 27\cdot 29^{5} + 3\cdot 29^{6} + 19\cdot 29^{7} + 12\cdot 29^{8} + 20\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 12 a^{2} + 5 a + 4 + \left(19 a^{2} + 15 a\right)\cdot 29 + \left(6 a^{2} + 3 a\right)\cdot 29^{2} + \left(27 a^{2} + 7 a + 15\right)\cdot 29^{3} + \left(28 a^{2} + 8 a + 18\right)\cdot 29^{4} + \left(6 a + 16\right)\cdot 29^{5} + \left(23 a^{2} + 25 a + 2\right)\cdot 29^{6} + \left(6 a^{2} + a + 25\right)\cdot 29^{7} + \left(12 a^{2} + 12 a\right)\cdot 29^{8} + \left(a^{2} + 9 a + 4\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.