Normalized defining polynomial
\( x^{8} - x^{7} + x^{6} - 4x^{5} + 5x^{4} - 8x^{3} + 4x^{2} - 8x + 16 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(705911761\) \(\medspace = 163^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.77\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $163^{2/3}\approx 29.839513369337443$ | ||
Ramified primes: | \(163\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | 8.0.705911761.1$^{8}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{2}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{5}{8}a^{7}+\frac{3}{8}a^{6}+\frac{9}{8}a^{5}-a^{4}+\frac{13}{8}a^{3}-3a^{2}-2a-8$, $\frac{1}{8}a^{7}+\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{3}{8}a^{3}-\frac{3}{4}a^{2}-\frac{3}{2}a-2$, $\frac{1}{8}a^{7}-\frac{3}{8}a^{6}+\frac{3}{8}a^{5}-\frac{3}{4}a^{4}+\frac{13}{8}a^{3}-\frac{5}{4}a^{2}+\frac{1}{2}a-2$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 62.8368450498 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 62.8368450498 \cdot 1}{2\cdot\sqrt{705911761}}\cr\approx \mathstrut & 1.84301402680 \end{aligned}\]
Galois group
$\SL(2,3)$ (as 8T12):
A solvable group of order 24 |
The 7 conjugacy class representatives for $\SL(2,3)$ |
Character table for $\SL(2,3)$ |
Intermediate fields
4.4.26569.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(163\) | 163.2.0.1 | $x^{2} + 159 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
163.6.4.1 | $x^{6} + 477 x^{5} + 75849 x^{4} + 4021913 x^{3} + 229449 x^{2} + 12362361 x + 655078752$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.163.3t1.a.a | $1$ | $ 163 $ | 3.3.26569.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.163.3t1.a.b | $1$ | $ 163 $ | 3.3.26569.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
2.26569.24t7.b.a | $2$ | $ 163^{2}$ | 8.0.705911761.1 | $\SL(2,3)$ (as 8T12) | $-1$ | $-2$ | |
* | 2.163.8t12.a.a | $2$ | $ 163 $ | 8.0.705911761.1 | $\SL(2,3)$ (as 8T12) | $0$ | $-2$ |
* | 2.163.8t12.a.b | $2$ | $ 163 $ | 8.0.705911761.1 | $\SL(2,3)$ (as 8T12) | $0$ | $-2$ |
* | 3.26569.4t4.a.a | $3$ | $ 163^{2}$ | 4.4.26569.1 | $A_4$ (as 4T4) | $1$ | $3$ |