Properties

Label 4.184914603.12t34.b.a
Dimension $4$
Group $C_3^2:D_4$
Conductor $184914603$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:D_4$
Conductor: \(184914603\)\(\medspace = 3^{3} \cdot 2617^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.70659.1
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 6.0.70659.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: \( x^{2} + 63x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 + 65\cdot 67 + 17\cdot 67^{2} + 10\cdot 67^{3} + 51\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 43 a + 62 + \left(24 a + 11\right)\cdot 67 + \left(25 a + 64\right)\cdot 67^{2} + \left(5 a + 29\right)\cdot 67^{3} + \left(36 a + 26\right)\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 56 + 34\cdot 67 + 38\cdot 67^{2} + 10\cdot 67^{3} + 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 24 a + 33 + 42 a\cdot 67 + \left(41 a + 7\right)\cdot 67^{2} + \left(61 a + 26\right)\cdot 67^{3} + \left(30 a + 31\right)\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 15 a + 28 + \left(33 a + 52\right)\cdot 67 + \left(35 a + 15\right)\cdot 67^{2} + \left(35 a + 42\right)\cdot 67^{3} + \left(59 a + 44\right)\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 52 a + 21 + \left(33 a + 36\right)\cdot 67 + \left(31 a + 57\right)\cdot 67^{2} + \left(31 a + 14\right)\cdot 67^{3} + \left(7 a + 46\right)\cdot 67^{4} +O(67^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.