Properties

Label 3.2e2_5_443.6t11.2c1
Dimension 3
Group $S_4\times C_2$
Conductor $ 2^{2} \cdot 5 \cdot 443 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$3$
Group:$S_4\times C_2$
Conductor:$8860= 2^{2} \cdot 5 \cdot 443 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - 5 x^{4} + 6 x^{3} + 22 x^{2} - 8 x - 32 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4\times C_2$
Parity: Even
Determinant: 1.2e2_5_443.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 21 + 4\cdot 41^{2} + 40\cdot 41^{3} + 25\cdot 41^{4} + 23\cdot 41^{5} + 31\cdot 41^{6} + 41^{7} + 4\cdot 41^{8} +O\left(41^{ 9 }\right)$
$r_{ 2 }$ $=$ $ 17 a + 1 + \left(18 a + 18\right)\cdot 41 + \left(31 a + 6\right)\cdot 41^{2} + \left(33 a + 14\right)\cdot 41^{3} + \left(22 a + 38\right)\cdot 41^{4} + \left(8 a + 17\right)\cdot 41^{5} + \left(25 a + 30\right)\cdot 41^{6} + \left(35 a + 36\right)\cdot 41^{7} + \left(38 a + 28\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$
$r_{ 3 }$ $=$ $ 26 + 22\cdot 41 + 39\cdot 41^{2} + 31\cdot 41^{3} + 38\cdot 41^{4} + 20\cdot 41^{5} + 37\cdot 41^{6} + 23\cdot 41^{7} + 28\cdot 41^{8} +O\left(41^{ 9 }\right)$
$r_{ 4 }$ $=$ $ 28 a + 32 + \left(10 a + 10\right)\cdot 41 + \left(10 a + 26\right)\cdot 41^{2} + \left(27 a + 22\right)\cdot 41^{3} + \left(38 a + 31\right)\cdot 41^{4} + \left(10 a + 22\right)\cdot 41^{5} + \left(22 a + 37\right)\cdot 41^{6} + \left(38 a + 26\right)\cdot 41^{7} + \left(22 a + 1\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$
$r_{ 5 }$ $=$ $ 24 a + 11 + \left(22 a + 15\right)\cdot 41 + 9 a\cdot 41^{2} + \left(7 a + 2\right)\cdot 41^{3} + \left(18 a + 32\right)\cdot 41^{4} + \left(32 a + 20\right)\cdot 41^{5} + \left(15 a + 15\right)\cdot 41^{6} + \left(5 a + 36\right)\cdot 41^{7} + \left(2 a + 27\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$
$r_{ 6 }$ $=$ $ 13 a + 34 + \left(30 a + 14\right)\cdot 41 + \left(30 a + 5\right)\cdot 41^{2} + \left(13 a + 12\right)\cdot 41^{3} + \left(2 a + 38\right)\cdot 41^{4} + \left(30 a + 16\right)\cdot 41^{5} + \left(18 a + 11\right)\cdot 41^{6} + \left(2 a + 38\right)\cdot 41^{7} + \left(18 a + 31\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$3$
$1$$2$$(1,3)(2,5)(4,6)$$-3$
$3$$2$$(1,3)$$1$
$3$$2$$(1,3)(2,5)$$-1$
$6$$2$$(2,4)(5,6)$$1$
$6$$2$$(1,3)(2,4)(5,6)$$-1$
$8$$3$$(1,2,4)(3,5,6)$$0$
$6$$4$$(1,5,3,2)$$1$
$6$$4$$(1,3)(2,6,5,4)$$-1$
$8$$6$$(1,5,6,3,2,4)$$0$
The blue line marks the conjugacy class containing complex conjugation.