Normalized defining polynomial
\( x^{8} - 4x^{6} - 12x^{5} + 37x^{4} + 24x^{3} - 30x^{2} - 198x + 223 \)
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: | \(24098215696\) \(\medspace = 2^{4}\cdot 197^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.85\) | 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: | $2^{2/3}197^{1/2}\approx 22.280235493786417$ | ||
Ramified primes: | \(2\), \(197\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{26}a^{6}+\frac{9}{26}a^{5}+\frac{5}{13}a^{4}-\frac{5}{26}a^{3}-\frac{1}{13}a^{2}+\frac{3}{26}a+\frac{1}{26}$, $\frac{1}{3302}a^{7}+\frac{6}{1651}a^{6}-\frac{1003}{3302}a^{5}+\frac{779}{3302}a^{4}+\frac{1257}{3302}a^{3}+\frac{1011}{3302}a^{2}+\frac{590}{1651}a+\frac{627}{3302}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{24}{1651}a^{7}+\frac{34}{1651}a^{6}+\frac{58}{1651}a^{5}-\frac{354}{1651}a^{4}+\frac{69}{1651}a^{3}+\frac{7}{1651}a^{2}+\frac{2793}{1651}a-\frac{259}{127}$, $\frac{75}{3302}a^{7}+\frac{11}{3302}a^{6}-\frac{26}{127}a^{5}+\frac{5}{3302}a^{4}+\frac{1481}{1651}a^{3}+\frac{1657}{3302}a^{2}-\frac{13227}{3302}a+\frac{4907}{1651}$, $2a^{4}-4a^{2}-12a+47$ | 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: | \( 112.729683635 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 112.729683635 \cdot 1}{2\cdot\sqrt{24098215696}}\cr\approx \mathstrut & 0.565894303663 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{197}) \), 4.0.788.1 |
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: | 4.0.788.1 |
Degree 6 siblings: | 6.2.122325968.1, 6.2.620944.1 |
Degree 12 siblings: | 12.4.14963642447137024.1, 12.0.75957575873792.1 |
Minimal sibling: | 4.0.788.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(197\) | 197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |