Normalized defining polynomial
\( x^{8} + 22x^{6} - 12x^{5} + 290x^{4} - 132x^{3} + 766x^{2} - 4401x + 4600 \)
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: | \(1624709678881\) \(\medspace = 1129^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.60\) | 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: | $1129^{1/2}\approx 33.60059523282288$ | ||
Ramified primes: | \(1129\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{9}a^{6}+\frac{1}{3}a^{5}+\frac{2}{9}a^{4}-\frac{1}{3}a^{3}-\frac{2}{9}a^{2}+\frac{1}{3}a-\frac{4}{9}$, $\frac{1}{15893527149}a^{7}+\frac{216569303}{15893527149}a^{6}-\frac{1824922873}{15893527149}a^{5}-\frac{1178713901}{15893527149}a^{4}+\frac{7884439210}{15893527149}a^{3}-\frac{5982212698}{15893527149}a^{2}+\frac{7216534964}{15893527149}a-\frac{259151141}{15893527149}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{9}$, which has order $9$
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: | $\frac{16171408}{15893527149}a^{7}+\frac{2504278}{1765947461}a^{6}+\frac{370420103}{15893527149}a^{5}+\frac{46982246}{1765947461}a^{4}+\frac{4701798853}{15893527149}a^{3}+\frac{785085274}{1765947461}a^{2}+\frac{12899614586}{15893527149}a-\frac{10610060249}{5297842383}$, $\frac{2543354}{5297842383}a^{7}-\frac{446862517}{15893527149}a^{6}+\frac{231810190}{1765947461}a^{5}-\frac{12894404225}{15893527149}a^{4}+\frac{4216643505}{1765947461}a^{3}-\frac{137354614375}{15893527149}a^{2}+\frac{29830383328}{1765947461}a-\frac{186261812603}{15893527149}$, $\frac{80}{1973}a^{7}-\frac{120}{1973}a^{6}+\frac{1940}{1973}a^{5}-\frac{3870}{1973}a^{4}+\frac{19140}{1973}a^{3}-\frac{39270}{1973}a^{2}+\frac{11670}{1973}a+\frac{21069}{1973}$ | 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: | \( 510.995413885 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 510.995413885 \cdot 9}{2\cdot\sqrt{1624709678881}}\cr\approx \mathstrut & 2.81164901554 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{1129}) \), 4.0.1129.1 |
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: | 4.0.1129.1 |
Degree 6 siblings: | 6.2.1274641.1, 6.2.1439069689.1 |
Degree 12 siblings: | deg 12, deg 12 |
Minimal sibling: | 4.0.1129.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(1129\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $4$ | $2$ | $2$ | $2$ |