Properties

Label 3.51681125.6t11.b.a
Dimension 3
Group $S_4\times C_2$
Conductor $ 5^{3} \cdot 643^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$S_4\times C_2$
Conductor:$51681125= 5^{3} \cdot 643^{2} $
Artin number field: Splitting field of 6.0.33230963375.2 defined by $f= x^{6} - 3 x^{5} + 44 x^{4} - 38 x^{3} + 650 x^{2} - 139 x + 5489 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4\times C_2$
Parity: Even
Determinant: 1.5.2t1.a.a
Projective image: $S_4$
Projective field: Galois closure of 4.2.643.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 12 a + 30 + \left(33 a + 10\right)\cdot 37 + \left(10 a + 25\right)\cdot 37^{2} + \left(30 a + 24\right)\cdot 37^{3} + \left(8 a + 16\right)\cdot 37^{4} + \left(26 a + 13\right)\cdot 37^{5} + \left(17 a + 34\right)\cdot 37^{6} + \left(34 a + 1\right)\cdot 37^{7} + \left(36 a + 22\right)\cdot 37^{8} + \left(11 a + 32\right)\cdot 37^{9} + \left(8 a + 4\right)\cdot 37^{10} + \left(7 a + 32\right)\cdot 37^{11} + \left(7 a + 11\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$
$r_{ 2 }$ $=$ $ 19 a + 10 + \left(18 a + 2\right)\cdot 37 + \left(22 a + 17\right)\cdot 37^{2} + \left(31 a + 11\right)\cdot 37^{3} + \left(20 a + 24\right)\cdot 37^{4} + \left(22 a + 22\right)\cdot 37^{5} + \left(34 a + 28\right)\cdot 37^{6} + \left(4 a + 18\right)\cdot 37^{7} + 30\cdot 37^{8} + \left(8 a + 7\right)\cdot 37^{9} + \left(3 a + 27\right)\cdot 37^{10} + \left(5 a + 33\right)\cdot 37^{11} + \left(28 a + 14\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$
$r_{ 3 }$ $=$ $ 36 + 18\cdot 37 + 23\cdot 37^{2} + 7\cdot 37^{4} + 25\cdot 37^{5} + 7\cdot 37^{6} + 2\cdot 37^{7} + 3\cdot 37^{8} + 33\cdot 37^{9} + 8\cdot 37^{10} + 27\cdot 37^{11} + 22\cdot 37^{12} +O\left(37^{ 13 }\right)$
$r_{ 4 }$ $=$ $ 22 + 32\cdot 37^{2} + 8\cdot 37^{3} + 2\cdot 37^{4} + 33\cdot 37^{5} + 37^{6} + 36\cdot 37^{7} + 4\cdot 37^{8} + 28\cdot 37^{9} + 11\cdot 37^{10} + 25\cdot 37^{11} + 16\cdot 37^{12} +O\left(37^{ 13 }\right)$
$r_{ 5 }$ $=$ $ 18 a + 12 + \left(18 a + 20\right)\cdot 37 + \left(14 a + 14\right)\cdot 37^{2} + \left(5 a + 4\right)\cdot 37^{3} + \left(16 a + 2\right)\cdot 37^{4} + \left(14 a + 18\right)\cdot 37^{5} + \left(2 a + 33\right)\cdot 37^{6} + \left(32 a + 3\right)\cdot 37^{7} + \left(36 a + 26\right)\cdot 37^{8} + \left(28 a + 2\right)\cdot 37^{9} + \left(33 a + 32\right)\cdot 37^{10} + \left(31 a + 13\right)\cdot 37^{11} + \left(8 a + 11\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$
$r_{ 6 }$ $=$ $ 25 a + 4 + \left(3 a + 21\right)\cdot 37 + \left(26 a + 35\right)\cdot 37^{2} + \left(6 a + 23\right)\cdot 37^{3} + \left(28 a + 21\right)\cdot 37^{4} + \left(10 a + 35\right)\cdot 37^{5} + \left(19 a + 4\right)\cdot 37^{6} + \left(2 a + 11\right)\cdot 37^{7} + 24\cdot 37^{8} + \left(25 a + 6\right)\cdot 37^{9} + \left(28 a + 26\right)\cdot 37^{10} + \left(29 a + 15\right)\cdot 37^{11} + \left(29 a + 33\right)\cdot 37^{12} +O\left(37^{ 13 }\right)$

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

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

Character values on conjugacy classes

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