Properties

Label 7.849...928.24t283.b.b
Dimension $7$
Group $C_2^3:(C_7: C_3)$
Conductor $8.491\times 10^{12}$
Root number not computed
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $7$
Group: $C_2^3:(C_7: C_3)$
Conductor: \(8491301244928\)\(\medspace = 2^{12} \cdot 73^{5} \)
Artin stem field: Galois closure of 8.0.116319195136.3
Galois orbit size: $2$
Smallest permutation container: 24T283
Parity: even
Determinant: 1.73.3t1.a.b
Projective image: $F_8:C_3$
Projective stem field: Galois closure of 8.0.116319195136.3

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{7} + 10x^{6} - 6x^{5} + 2x^{4} + 8x^{3} - 16x^{2} + 16x + 26 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{3} + 2x + 9 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a^{2} + 9 a + 10 + \left(7 a + 8\right)\cdot 11 + \left(8 a^{2} + 4\right)\cdot 11^{2} + \left(6 a^{2} + 7 a + 4\right)\cdot 11^{3} + \left(5 a^{2} + 6 a + 2\right)\cdot 11^{4} + \left(7 a^{2} + 9 a + 9\right)\cdot 11^{5} + \left(9 a^{2} + a + 10\right)\cdot 11^{6} + \left(9 a^{2} + 2 a\right)\cdot 11^{7} + \left(9 a^{2} + 2 a + 3\right)\cdot 11^{8} + \left(9 a^{2} + 5 a + 10\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 4 + 4\cdot 11 + 7\cdot 11^{2} + 4\cdot 11^{4} + 6\cdot 11^{5} + 9\cdot 11^{6} + 9\cdot 11^{7} + 11^{8} +O(11^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 a + 8 + \left(a^{2} + 6\right)\cdot 11 + \left(5 a^{2} + 7 a + 3\right)\cdot 11^{2} + \left(6 a^{2} + 5 a + 6\right)\cdot 11^{3} + \left(a^{2} + 4 a + 3\right)\cdot 11^{4} + \left(5 a^{2} + 9 a + 4\right)\cdot 11^{5} + \left(2 a^{2} + 6 a + 1\right)\cdot 11^{6} + \left(a^{2} + 8 a + 10\right)\cdot 11^{7} + \left(5 a^{2} + 5 a + 1\right)\cdot 11^{8} + \left(10 a^{2} + 6 a + 10\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a^{2} + 7 a + 9 + \left(10 a^{2} + 2 a + 7\right)\cdot 11 + \left(2 a^{2} + 6 a + 1\right)\cdot 11^{2} + \left(6 a^{2} + 4 a\right)\cdot 11^{3} + \left(7 a^{2} + 4 a + 5\right)\cdot 11^{4} + \left(2 a^{2} + a + 6\right)\cdot 11^{5} + \left(3 a^{2} + 9\right)\cdot 11^{6} + \left(a^{2} + 6 a + 7\right)\cdot 11^{7} + \left(6 a^{2} + 1\right)\cdot 11^{8} + \left(4 a^{2} + a + 3\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a^{2} + 6 a + 1 + \left(9 a^{2} + 4 a + 3\right)\cdot 11 + \left(3 a^{2} + 4 a + 9\right)\cdot 11^{2} + \left(6 a^{2} + 2 a + 9\right)\cdot 11^{3} + \left(3 a^{2} + 8 a + 9\right)\cdot 11^{4} + \left(5 a^{2} + a\right)\cdot 11^{5} + \left(4 a^{2} + 3 a + 4\right)\cdot 11^{6} + \left(9 a^{2} + 5 a + 6\right)\cdot 11^{7} + \left(10 a^{2} + 4 a + 9\right)\cdot 11^{8} + \left(2 a^{2} + 8 a + 3\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 7 a^{2} + 6 a + 7 + 9\cdot 11 + \left(4 a + 8\right)\cdot 11^{2} + \left(9 a^{2} + 10 a + 3\right)\cdot 11^{3} + \left(8 a^{2} + 10 a + 10\right)\cdot 11^{4} + \left(10 a + 3\right)\cdot 11^{5} + \left(9 a^{2} + 8 a + 6\right)\cdot 11^{6} + \left(10 a^{2} + 2 a + 9\right)\cdot 11^{7} + \left(5 a^{2} + 8 a + 8\right)\cdot 11^{8} + \left(7 a^{2} + 4 a + 10\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 8 a^{2} + 8 a + 4 + \left(5 a + 6\right)\cdot 11 + \left(2 a^{2} + 10 a + 10\right)\cdot 11^{2} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{3} + \left(5 a^{2} + 9 a + 1\right)\cdot 11^{4} + \left(10 a + 9\right)\cdot 11^{5} + \left(4 a^{2} + 10\right)\cdot 11^{6} + \left(8 a + 8\right)\cdot 11^{7} + \left(6 a^{2} + 6\right)\cdot 11^{8} + \left(8 a^{2} + 7 a + 7\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 3 + 8\cdot 11 + 8\cdot 11^{2} + 8\cdot 11^{3} + 6\cdot 11^{4} + 3\cdot 11^{5} + 2\cdot 11^{6} + 11^{7} + 10\cdot 11^{8} + 8\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$7$
$7$$2$$(1,3)(2,8)(4,5)(6,7)$$-1$
$28$$3$$(1,4,2)(3,5,8)$$-\zeta_{3} - 1$
$28$$3$$(1,2,4)(3,8,5)$$\zeta_{3}$
$28$$6$$(1,5,2,3,4,8)(6,7)$$\zeta_{3} + 1$
$28$$6$$(1,8,4,3,2,5)(6,7)$$-\zeta_{3}$
$24$$7$$(2,5,6,4,3,7,8)$$0$
$24$$7$$(2,4,8,6,7,5,3)$$0$

The blue line marks the conjugacy class containing complex conjugation.