Normalized defining polynomial
\( x^{8} - 2x^{7} - 13x^{6} + 13x^{5} + 193x^{4} - 136x^{3} - 1247x^{2} + 421x + 4866 \)
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: | \(941336550625\) \(\medspace = 5^{4}\cdot 197^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(31.38\) | 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: | $5^{1/2}197^{1/2}\approx 31.38470965295043$ | ||
Ramified primes: | \(5\), \(197\) | 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 not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{16}a^{6}+\frac{1}{8}a^{4}+\frac{1}{16}a^{3}+\frac{1}{16}a^{2}-\frac{7}{16}a+\frac{1}{8}$, $\frac{1}{37880384}a^{7}-\frac{266585}{37880384}a^{6}+\frac{1714329}{18940192}a^{5}-\frac{6150065}{37880384}a^{4}+\frac{234773}{4735048}a^{3}+\frac{343447}{1183762}a^{2}-\frac{7792383}{37880384}a-\frac{2770233}{18940192}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{6}$, which has order $12$
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: | $\frac{1361}{9470096}a^{7}+\frac{217}{2367524}a^{6}+\frac{6867}{4735048}a^{5}+\frac{142637}{9470096}a^{4}-\frac{125615}{9470096}a^{3}-\frac{1988531}{9470096}a^{2}+\frac{833069}{4735048}a+\frac{517253}{591881}$, $\frac{17459}{37880384}a^{7}+\frac{244669}{37880384}a^{6}+\frac{231603}{18940192}a^{5}-\frac{2036387}{37880384}a^{4}-\frac{132235}{591881}a^{3}+\frac{1354691}{4735048}a^{2}+\frac{80680155}{37880384}a+\frac{50367853}{18940192}$, $\frac{13}{32}a^{7}-\frac{13}{32}a^{6}-\frac{91}{16}a^{5}-\frac{13}{32}a^{4}+52a^{3}+\frac{91}{4}a^{2}-\frac{9659}{32}a+\frac{4643}{16}$ | 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: | \( 270.609787768 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 270.609787768 \cdot 12}{2\cdot\sqrt{941336550625}}\cr\approx \mathstrut & 2.60820524243 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{985}) \), 4.0.985.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 4 sibling: | 4.0.985.1 |
Degree 6 siblings: | 6.2.955671625.1, 6.2.970225.1 |
Degree 12 siblings: | deg 12, deg 12 |
Minimal sibling: | 4.0.985.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. 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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(197\) | 197.2.1.2 | $x^{2} + 394$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
197.2.1.2 | $x^{2} + 394$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
197.2.1.2 | $x^{2} + 394$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
197.2.1.2 | $x^{2} + 394$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |