Properties

Label 3.2527.6t6.a.a
Dimension $3$
Group $A_4\times C_2$
Conductor $2527$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_4\times C_2$
Conductor: \(2527\)\(\medspace = 7 \cdot 19^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.4.912247.1
Galois orbit size: $1$
Smallest permutation container: $A_4\times C_2$
Parity: odd
Determinant: 1.7.2t1.a.a
Projective image: $A_4$
Projective stem field: Galois closure of 4.0.17689.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} - 3x^{4} + 5x^{3} - 3x^{2} - x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.