Properties

Label 4.4232000.6t10.b.a
Dimension $4$
Group $C_3^2:C_4$
Conductor $4232000$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:C_4$
Conductor: \(4232000\)\(\medspace = 2^{6} \cdot 5^{3} \cdot 23^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.21160000.3
Galois orbit size: $1$
Smallest permutation container: $C_3^2:C_4$
Parity: even
Determinant: 1.5.2t1.a.a
Projective image: $C_3^2:C_4$
Projective stem field: Galois closure of 6.2.21160000.3

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + x^{4} - 20x^{3} + 15x^{2} + 30x + 55 \) 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 $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \) Copy content Toggle raw display

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

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$9$$2$$(1,2)(4,5)$$0$
$4$$3$$(1,2,3)$$-2$
$4$$3$$(1,2,3)(4,5,6)$$1$
$9$$4$$(1,5,2,4)(3,6)$$0$
$9$$4$$(1,4,2,5)(3,6)$$0$

The blue line marks the conjugacy class containing complex conjugation.