Properties

Label 2.2023.6t3.a.a
Dimension $2$
Group $D_{6}$
Conductor $2023$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(2023\)\(\medspace = 7 \cdot 17^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.69572993.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.7.2t1.a.a
Projective image: S_3
Projective stem field: 3.1.2023.1

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} - x^{4} + 21 x^{3} + 25 x^{2} - 13 x - 19\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{2} + 12 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 1 + 2\cdot 13 + 11\cdot 13^{2} + 13^{3} + 11\cdot 13^{4} + 7\cdot 13^{6} + 10\cdot 13^{7} +O(13^{8})\)  Toggle raw display
$r_{ 2 }$ $=$ \( a + \left(6 a + 11\right)\cdot 13 + \left(5 a + 7\right)\cdot 13^{2} + \left(12 a + 12\right)\cdot 13^{3} + \left(6 a + 10\right)\cdot 13^{4} + 8 a\cdot 13^{5} + \left(3 a + 9\right)\cdot 13^{6} + \left(6 a + 2\right)\cdot 13^{7} +O(13^{8})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 12 a + 1 + \left(6 a + 3\right)\cdot 13 + \left(7 a + 7\right)\cdot 13^{2} + 6\cdot 13^{3} + \left(6 a + 5\right)\cdot 13^{4} + \left(4 a + 2\right)\cdot 13^{5} + \left(9 a + 4\right)\cdot 13^{6} + \left(6 a + 5\right)\cdot 13^{7} +O(13^{8})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 10 a + 5 + 5 a\cdot 13 + \left(4 a + 1\right)\cdot 13^{2} + \left(2 a + 9\right)\cdot 13^{3} + \left(5 a + 12\right)\cdot 13^{4} + \left(3 a + 12\right)\cdot 13^{5} + \left(10 a + 8\right)\cdot 13^{6} + \left(9 a + 1\right)\cdot 13^{7} +O(13^{8})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 3 a + 2 + \left(7 a + 9\right)\cdot 13 + \left(8 a + 12\right)\cdot 13^{2} + \left(10 a + 6\right)\cdot 13^{3} + \left(7 a + 2\right)\cdot 13^{4} + \left(9 a + 11\right)\cdot 13^{5} + \left(2 a + 2\right)\cdot 13^{6} + \left(3 a + 1\right)\cdot 13^{7} +O(13^{8})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 5 + 12\cdot 13^{2} + 13^{3} + 9\cdot 13^{4} + 10\cdot 13^{5} + 6\cdot 13^{6} + 4\cdot 13^{7} +O(13^{8})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.