Properties

Label 3.41e3_257e2.6t11.2c1
Dimension 3
Group $S_4\times C_2$
Conductor $ 41^{3} \cdot 257^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$3$
Group:$S_4\times C_2$
Conductor:$4552163129= 41^{3} \cdot 257^{2} $
Artin number field: Splitting field of $f=x^{6} - 2 x^{5} + 18 x^{4} + 56 x^{3} - 257 x^{2} - 794 x - 1671$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4\times C_2$
Parity: Even
Determinant: 1.41.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $x^{2} + 60 x + 2$
Roots: \[ \begin{aligned} r_{ 1 } &= -63476863443391997279 +O\left(61^{ 12 }\right) \\ r_{ 2 } &= -400410767552906194475 a - 67483561892818165132 +O\left(61^{ 12 }\right) \\ r_{ 3 } &= 1075762725181747628611 a - 1212099706827032726355 +O\left(61^{ 12 }\right) \\ r_{ 4 } &= 674177269852201295279 +O\left(61^{ 12 }\right) \\ r_{ 5 } &= -1075762725181747628611 a + 600861139707763515010 +O\left(61^{ 12 }\right) \\ r_{ 6 } &= 400410767552906194475 a + 68021722603278078479 +O\left(61^{ 12 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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