Properties

Label 4.3e2_7_127.6t13.2c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 3^{2} \cdot 7 \cdot 127 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$8001= 3^{2} \cdot 7 \cdot 127 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 3 x^{3} - x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.7_127.2t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,2)(3,5)(4,6)$$0$
$6$$2$$(3,4)$$2$
$9$$2$$(3,4)(5,6)$$0$
$4$$3$$(1,3,4)(2,5,6)$$-2$
$4$$3$$(1,3,4)$$1$
$18$$4$$(1,2)(3,6,4,5)$$0$
$12$$6$$(1,5,3,6,4,2)$$0$
$12$$6$$(2,5,6)(3,4)$$-1$
The blue line marks the conjugacy class containing complex conjugation.