Properties

Label 2.3311.6t3.f.a
Dimension $2$
Group $D_{6}$
Conductor $3311$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(3311\)\(\medspace = 7 \cdot 11 \cdot 43 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.471397003.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.3311.2t1.a.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.3311.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} - 55x^{2} + 64x + 97 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 21 a + 1 + 30 a\cdot 31 + \left(15 a + 22\right)\cdot 31^{2} + \left(20 a + 11\right)\cdot 31^{3} + \left(12 a + 14\right)\cdot 31^{4} + \left(25 a + 25\right)\cdot 31^{5} + \left(4 a + 7\right)\cdot 31^{6} +O(31^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 16\cdot 31 + 2\cdot 31^{2} + 23\cdot 31^{3} + 30\cdot 31^{4} + 20\cdot 31^{5} + 15\cdot 31^{6} +O(31^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 24 + 14\cdot 31 + 28\cdot 31^{2} + 7\cdot 31^{3} + 10\cdot 31^{5} + 15\cdot 31^{6} +O(31^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 10 a + \left(15 a + 9\right)\cdot 31^{2} + \left(10 a + 19\right)\cdot 31^{3} + \left(18 a + 16\right)\cdot 31^{4} + \left(5 a + 5\right)\cdot 31^{5} + \left(26 a + 23\right)\cdot 31^{6} +O(31^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 21 a + 20 + \left(30 a + 21\right)\cdot 31 + \left(15 a + 7\right)\cdot 31^{2} + \left(20 a + 25\right)\cdot 31^{3} + \left(12 a + 11\right)\cdot 31^{4} + \left(25 a + 29\right)\cdot 31^{5} + \left(4 a + 7\right)\cdot 31^{6} +O(31^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 10 a + 12 + 9\cdot 31 + \left(15 a + 23\right)\cdot 31^{2} + \left(10 a + 5\right)\cdot 31^{3} + \left(18 a + 19\right)\cdot 31^{4} + \left(5 a + 1\right)\cdot 31^{5} + \left(26 a + 23\right)\cdot 31^{6} +O(31^{7})\) Copy content Toggle raw display

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

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

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$$2$$(1,2)(3,4)(5,6)$$0$
$3$$2$$(1,6)(4,5)$$0$
$2$$3$$(1,3,6)(2,5,4)$$-1$
$2$$6$$(1,5,3,4,6,2)$$1$

The blue line marks the conjugacy class containing complex conjugation.