# Properties

 Label 4.2e6_3e3_5e3_17e3.8t39.17c1 Dimension 4 Group $C_2^3:S_4$ Conductor $2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 17^{3}$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $4$ Group: $C_2^3:S_4$ Conductor: $1061208000= 2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 17^{3}$ Artin number field: Splitting field of $f=x^{8} + 8 x^{6} - 40 x^{2} + 100$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 8T39 Parity: Odd Determinant: 1.3_5_17.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 36.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $x^{3} + 2 x + 9$
Roots: \begin{aligned} r_{ 1 } &= -8381525186661357120154968651896189888 +O\left(11^{ 36 }\right) \\ r_{ 2 } &= 5404892482196723666509330499430332553 a^{2} - 11376198584060197061513167420436659457 a - 14735260895293368261978043236862442266 +O\left(11^{ 36 }\right) \\ r_{ 3 } &= -15039052254470478905695727149637403145 a^{2} - 3930917365201666479830126075793089040 a - 776946500355408177353649519769485382 +O\left(11^{ 36 }\right) \\ r_{ 4 } &= 9634159772273755239186396650207070592 a^{2} + 15307115949261863541343293496229748497 a + 11512215846723122258706946922098177360 +O\left(11^{ 36 }\right) \\ r_{ 5 } &= 8381525186661357120154968651896189888 +O\left(11^{ 36 }\right) \\ r_{ 6 } &= -5404892482196723666509330499430332553 a^{2} + 11376198584060197061513167420436659457 a + 14735260895293368261978043236862442266 +O\left(11^{ 36 }\right) \\ r_{ 7 } &= 15039052254470478905695727149637403145 a^{2} + 3930917365201666479830126075793089040 a + 776946500355408177353649519769485382 +O\left(11^{ 36 }\right) \\ r_{ 8 } &= -9634159772273755239186396650207070592 a^{2} - 15307115949261863541343293496229748497 a - 11512215846723122258706946922098177360 +O\left(11^{ 36 }\right) \\ \end{aligned}

### Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

 Cycle notation $(2,8)(4,6)$ $(1,3,6,4)(2,8,5,7)$ $(2,4)(6,8)$ $(1,3,2,4)(5,7,6,8)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $4$ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$ $-4$ $6$ $2$ $(1,6)(2,5)(3,4)(7,8)$ $0$ $6$ $2$ $(1,6)(2,5)(3,8)(4,7)$ $0$ $6$ $2$ $(3,7)(4,8)$ $0$ $12$ $2$ $(2,8)(4,6)$ $-2$ $12$ $2$ $(1,5)(2,6)(3,8)(4,7)$ $2$ $32$ $3$ $(2,4,3)(6,8,7)$ $1$ $12$ $4$ $(1,2,5,6)(3,8,7,4)$ $0$ $24$ $4$ $(1,3,6,4)(2,8,5,7)$ $0$ $24$ $4$ $(1,7,6,4)(2,8,5,3)$ $0$ $24$ $4$ $(1,5)(3,8,7,4)$ $0$ $32$ $6$ $(1,5)(2,7,4,6,3,8)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.