Properties

Label 2.4356.6t3.d.a
Dimension $2$
Group $D_{6}$
Conductor $4356$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(4356\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.25299648.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.484.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{4} - 22x^{3} - 29x^{2} - 22x + 46 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 21\cdot 23 + 21\cdot 23^{2} + 12\cdot 23^{3} + 8\cdot 23^{4} + 19\cdot 23^{5} + 3\cdot 23^{6} +O(23^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 a + 19 + \left(12 a + 9\right)\cdot 23 + \left(7 a + 4\right)\cdot 23^{2} + \left(16 a + 3\right)\cdot 23^{3} + \left(18 a + 5\right)\cdot 23^{4} + \left(19 a + 16\right)\cdot 23^{5} + \left(15 a + 6\right)\cdot 23^{6} +O(23^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 6 a + 9 + \left(14 a + 19\right)\cdot 23 + \left(3 a + 21\right)\cdot 23^{2} + \left(3 a + 2\right)\cdot 23^{3} + \left(4 a + 18\right)\cdot 23^{4} + \left(20 a + 21\right)\cdot 23^{5} + \left(4 a + 16\right)\cdot 23^{6} +O(23^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 18 + 14\cdot 23 + 21\cdot 23^{2} + 15\cdot 23^{3} + 10\cdot 23^{4} + 8\cdot 23^{5} + 16\cdot 23^{6} +O(23^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 17 a + 21 + \left(8 a + 18\right)\cdot 23 + \left(19 a + 14\right)\cdot 23^{2} + \left(19 a + 5\right)\cdot 23^{3} + 18 a\cdot 23^{4} + \left(2 a + 12\right)\cdot 23^{5} + \left(18 a + 6\right)\cdot 23^{6} +O(23^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 20 a + 2 + \left(10 a + 8\right)\cdot 23 + \left(15 a + 7\right)\cdot 23^{2} + \left(6 a + 5\right)\cdot 23^{3} + \left(4 a + 3\right)\cdot 23^{4} + \left(3 a + 14\right)\cdot 23^{5} + \left(7 a + 18\right)\cdot 23^{6} +O(23^{7})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.