Properties

Label 4.28053493.12t34.b.a
Dimension $4$
Group $C_3^2:D_4$
Conductor $28053493$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:D_4$
Conductor: \(28053493\)\(\medspace = 13^{3} \cdot 113^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.248261.1
Galois orbit size: $1$
Smallest permutation container: 12T34
Parity: even
Determinant: 1.13.2t1.a.a
Projective image: $\SOPlus(4,2)$
Projective stem field: Galois closure of 6.2.248261.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: \( x^{2} + 60x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 35 + 19\cdot 61 + 2\cdot 61^{2} + 2\cdot 61^{3} + 33\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 a + 47 + \left(44 a + 16\right)\cdot 61 + \left(a + 56\right)\cdot 61^{2} + \left(8 a + 15\right)\cdot 61^{3} + \left(35 a + 57\right)\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 43 a + 11 + \left(30 a + 24\right)\cdot 61 + \left(48 a + 56\right)\cdot 61^{2} + \left(57 a + 50\right)\cdot 61^{3} + \left(14 a + 5\right)\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 18 a + 54 + \left(30 a + 11\right)\cdot 61 + \left(12 a + 13\right)\cdot 61^{2} + \left(3 a + 60\right)\cdot 61^{3} + \left(46 a + 23\right)\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 41 + 41\cdot 61^{2} + 31\cdot 61^{3} + 39\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 49 a + 59 + \left(16 a + 48\right)\cdot 61 + \left(59 a + 13\right)\cdot 61^{2} + \left(52 a + 22\right)\cdot 61^{3} + \left(25 a + 23\right)\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.