Normalized defining polynomial
\( x^{9} - 3x^{8} + 3x^{7} - 15x^{6} + 33x^{5} - 3x^{4} + 24x^{3} + 6x^{2} - 4 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2125764000000\) \(\medspace = 2^{8}\cdot 3^{12}\cdot 5^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(23.43\) | 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))
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Galois root discriminant: | $2\cdot 3^{25/18}5^{3/4}\approx 30.755720695068224$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{6}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{50}a^{7}+\frac{3}{50}a^{6}-\frac{3}{50}a^{5}-\frac{1}{2}a^{4}+\frac{3}{10}a^{3}+\frac{7}{50}a^{2}+\frac{8}{25}a+\frac{7}{25}$, $\frac{1}{50}a^{8}-\frac{1}{25}a^{6}+\frac{2}{25}a^{5}-\frac{2}{5}a^{4}-\frac{9}{25}a^{3}+\frac{1}{10}a^{2}+\frac{8}{25}a-\frac{6}{25}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{1}{10}a^{8}-\frac{7}{25}a^{7}+\frac{4}{25}a^{6}-\frac{29}{25}a^{5}+\frac{13}{5}a^{4}+\frac{9}{5}a^{3}-\frac{53}{50}a^{2}+\frac{68}{25}a-\frac{33}{25}$, $\frac{1}{5}a^{8}-\frac{43}{50}a^{7}+\frac{51}{50}a^{6}-\frac{131}{50}a^{5}+\frac{19}{2}a^{4}-\frac{9}{2}a^{3}-\frac{351}{50}a^{2}-\frac{114}{25}a-\frac{11}{25}$, $\frac{1}{25}a^{8}-\frac{11}{50}a^{7}+\frac{23}{50}a^{6}-\frac{39}{50}a^{5}+\frac{5}{2}a^{4}-\frac{181}{50}a^{3}-\frac{7}{50}a^{2}+\frac{28}{25}a+\frac{26}{25}$, $\frac{1}{50}a^{8}-\frac{44}{25}a^{7}+\frac{122}{25}a^{6}-\frac{76}{25}a^{5}+\frac{106}{5}a^{4}-\frac{1249}{25}a^{3}-\frac{1091}{50}a^{2}+\frac{69}{25}a+\frac{198}{25}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 1978.70362296 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 1978.70362296 \cdot 1}{2\cdot\sqrt{2125764000000}}\cr\approx \mathstrut & 2.11515743581 \end{aligned}\]
Galois group
$C_3^2:C_4$ (as 9T9):
A solvable group of order 36 |
The 6 conjugacy class representatives for $C_3^2:C_4$ |
Character table for $C_3^2:C_4$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | data not computed |
Degree 6 siblings: | 6.2.7290000.1, 6.2.7290000.2 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 18 sibling: | deg 18 |
Minimal sibling: | 6.2.7290000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
\(3\) | 3.9.12.21 | $x^{9} + 6 x^{4} + 3$ | $9$ | $1$ | $12$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |