Properties

Label 4.64783827.12t34.b.a
Dimension $4$
Group $C_3^2:D_4$
Conductor $64783827$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:D_4$
Conductor: \(64783827\)\(\medspace = 3^{3} \cdot 1549^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.41823.1
Galois orbit size: $1$
Smallest permutation container: 12T34
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $\SOPlus(4,2)$
Projective stem field: Galois closure of 6.0.41823.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \( x^{2} + 6x + 3 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,2)(3,4)(5,6)$$-2$
$6$$2$$(3,5)$$0$
$9$$2$$(3,5)(4,6)$$0$
$4$$3$$(1,4,6)$$-2$
$4$$3$$(1,4,6)(2,3,5)$$1$
$18$$4$$(1,2)(3,6,5,4)$$0$
$12$$6$$(1,3,4,5,6,2)$$1$
$12$$6$$(1,4,6)(3,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.