Properties

Label 4.8956368.6t13.a.a
Dimension $4$
Group $C_3^2:D_4$
Conductor $8956368$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $4$
Group: $C_3^2:D_4$
Conductor: \(8956368\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 37 \cdot 41^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.299055312.1
Galois orbit size: $1$
Smallest permutation container: $C_3^2:D_4$
Parity: even
Determinant: 1.37.2t1.a.a
Projective image: $S_3\wr C_2$
Projective stem field: Galois closure of 6.2.299055312.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} + 10x^{4} + 5x^{3} + 46x^{2} - x - 33 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: \( x^{2} + 70x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 23 a + 34 + \left(25 a + 51\right)\cdot 73 + \left(47 a + 27\right)\cdot 73^{2} + \left(46 a + 18\right)\cdot 73^{3} + \left(39 a + 50\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 60 a + 50 + \left(11 a + 60\right)\cdot 73 + \left(53 a + 36\right)\cdot 73^{2} + \left(6 a + 13\right)\cdot 73^{3} + \left(58 a + 10\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 a + 11 + \left(61 a + 36\right)\cdot 73 + \left(19 a + 38\right)\cdot 73^{2} + \left(66 a + 53\right)\cdot 73^{3} + \left(14 a + 31\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 50 a + 30 + \left(47 a + 31\right)\cdot 73 + \left(25 a + 71\right)\cdot 73^{2} + \left(26 a + 37\right)\cdot 73^{3} + \left(33 a + 49\right)\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 57 + 61\cdot 73 + 19\cdot 73^{2} + 67\cdot 73^{3} + 41\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 38 + 50\cdot 73 + 24\cdot 73^{2} + 28\cdot 73^{3} + 35\cdot 73^{4} +O(73^{5})\) 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$$2$$(1,4)$$0$
$9$$2$$(1,4)(2,3)$$0$
$4$$3$$(1,4,6)(2,3,5)$$1$
$4$$3$$(2,3,5)$$-2$
$18$$4$$(1,3,4,2)(5,6)$$0$
$12$$6$$(1,2,4,3,6,5)$$-1$
$12$$6$$(1,4)(2,3,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.