Properties

Label 7.243...233.24t283.a.a
Dimension $7$
Group $C_2^3:(C_7: C_3)$
Conductor $2.433\times 10^{14}$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $7$
Group: $C_2^3:(C_7: C_3)$
Conductor: \(243336191536233\)\(\medspace = 3^{4} \cdot 313^{5} \)
Artin stem field: Galois closure of 8.0.777431921841.1
Galois orbit size: $2$
Smallest permutation container: 24T283
Parity: even
Determinant: 1.313.3t1.a.a
Projective image: $F_8:C_3$
Projective stem field: Galois closure of 8.0.777431921841.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 9x^{6} - 4x^{5} + 24x^{4} + 21x^{3} + 22x^{2} - 3x + 3 \) 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:
$r_{ 1 }$ $=$ \( 1 + 3\cdot 11 + 3\cdot 11^{2} + 8\cdot 11^{3} + 5\cdot 11^{4} + 3\cdot 11^{5} + 11^{6} + 3\cdot 11^{7} + 3\cdot 11^{8} + 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 a^{2} + 6 a + 3 + \left(8 a^{2} + 7 a + 7\right)\cdot 11 + \left(2 a^{2} + 7\right)\cdot 11^{2} + \left(5 a + 4\right)\cdot 11^{3} + \left(9 a^{2} + 2 a + 1\right)\cdot 11^{4} + \left(2 a^{2} + 10 a + 5\right)\cdot 11^{5} + \left(2 a^{2} + 4 a + 10\right)\cdot 11^{6} + \left(9 a^{2} + 5 a + 1\right)\cdot 11^{7} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{8} + \left(10 a^{2} + 6 a + 6\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a^{2} + a + 6 + \left(2 a^{2} + a + 9\right)\cdot 11 + \left(8 a^{2} + 4 a + 3\right)\cdot 11^{2} + \left(2 a^{2} + 10 a + 4\right)\cdot 11^{3} + \left(4 a^{2} + 6 a + 2\right)\cdot 11^{4} + \left(4 a^{2} + 6 a + 7\right)\cdot 11^{5} + \left(7 a^{2} + 6 a + 2\right)\cdot 11^{6} + \left(9 a^{2} + 4 a + 6\right)\cdot 11^{7} + \left(a^{2} + 6 a + 9\right)\cdot 11^{8} + \left(a^{2} + 2 a + 4\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 9 + 10\cdot 11 + 3\cdot 11^{2} + 6\cdot 11^{3} + 8\cdot 11^{5} + 11^{6} + 8\cdot 11^{7} + 10\cdot 11^{8} + 8\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 10 a^{2} + 8 a + 3 + \left(a + 8\right)\cdot 11 + \left(9 a^{2} + 5\right)\cdot 11^{2} + \left(5 a^{2} + 2 a + 9\right)\cdot 11^{3} + \left(6 a^{2} + 3 a + 9\right)\cdot 11^{4} + 2\cdot 11^{5} + \left(6 a^{2} + 8 a + 3\right)\cdot 11^{6} + \left(9 a^{2} + 9 a + 8\right)\cdot 11^{7} + \left(a^{2} + 7 a + 1\right)\cdot 11^{8} + \left(3 a^{2} + 10 a + 1\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 10 a^{2} + 3 + \left(2 a^{2} + a + 7\right)\cdot 11 + \left(8 a^{2} + 3 a + 4\right)\cdot 11^{2} + \left(4 a^{2} + 7 a + 4\right)\cdot 11^{3} + \left(8 a^{2} + 1\right)\cdot 11^{4} + \left(3 a^{2} + 8 a + 7\right)\cdot 11^{5} + \left(6 a^{2} + 6 a + 3\right)\cdot 11^{6} + \left(a^{2} + 8 a + 1\right)\cdot 11^{7} + \left(4 a^{2} + 5 a + 1\right)\cdot 11^{8} + \left(4 a^{2} + 9 a + 10\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 2 a^{2} + 3 a + 7 + \left(7 a^{2} + 8 a + 5\right)\cdot 11 + \left(4 a^{2} + 7 a + 3\right)\cdot 11^{2} + \left(a + 2\right)\cdot 11^{3} + \left(7 a^{2} + 7 a + 3\right)\cdot 11^{4} + \left(6 a^{2} + 2 a + 7\right)\cdot 11^{5} + \left(9 a^{2} + 7 a\right)\cdot 11^{6} + \left(10 a^{2} + 3 a + 10\right)\cdot 11^{7} + \left(4 a^{2} + 8 a + 5\right)\cdot 11^{8} + \left(3 a^{2} + a + 1\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( a^{2} + 4 a + 1 + \left(2 a + 3\right)\cdot 11 + 6 a\cdot 11^{2} + \left(8 a^{2} + 6 a + 4\right)\cdot 11^{3} + \left(8 a^{2} + a + 8\right)\cdot 11^{4} + \left(3 a^{2} + 5 a + 2\right)\cdot 11^{5} + \left(a^{2} + 10 a + 9\right)\cdot 11^{6} + \left(3 a^{2} + 4\right)\cdot 11^{7} + \left(10 a^{2} + 2 a + 2\right)\cdot 11^{8} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.