Properties

Label 2.41e2_257.6t3.2c1
Dimension 2
Group $D_{6}$
Conductor $ 41^{2} \cdot 257 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:$432017= 41^{2} \cdot 257 $
Artin number field: Splitting field of $f=x^{6} - 2 x^{5} - 267 x^{4} + 504 x^{3} + 17720 x^{2} - 31624 x - 249501$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{6}$
Parity: Even
Determinant: 1.257.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$
Roots: \[ \begin{aligned} r_{ 1 } &= -488519553380745 a + 388251348590356 +O\left(47^{ 9 }\right) \\ r_{ 2 } &= -290687329545959 a + 133242669077364 +O\left(47^{ 9 }\right) \\ r_{ 3 } &= -377273137258896 +O\left(47^{ 9 }\right) \\ r_{ 4 } &= 488519553380745 a - 10978211331459 +O\left(47^{ 9 }\right) \\ r_{ 5 } &= 290687329545959 a - 215393178587685 +O\left(47^{ 9 }\right) \\ r_{ 6 } &= 82150509510322 +O\left(47^{ 9 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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