Normalized defining polynomial
\( x^{18} + 3x^{12} + 15x^{6} + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-2518170116818978404827136\) \(\medspace = -\,2^{24}\cdot 3^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(22.68\) | 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: | $2^{4/3}3^{37/18}\approx 24.105856857882777$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{10}a^{12}-\frac{2}{5}a^{6}+\frac{3}{10}$, $\frac{1}{10}a^{13}-\frac{2}{5}a^{7}+\frac{3}{10}a$, $\frac{1}{10}a^{14}-\frac{2}{5}a^{8}+\frac{3}{10}a^{2}$, $\frac{1}{10}a^{15}+\frac{1}{10}a^{9}-\frac{1}{2}a^{6}-\frac{1}{5}a^{3}-\frac{1}{2}$, $\frac{1}{10}a^{16}+\frac{1}{10}a^{10}-\frac{1}{2}a^{7}-\frac{1}{5}a^{4}-\frac{1}{2}a$, $\frac{1}{10}a^{17}+\frac{1}{10}a^{11}-\frac{1}{2}a^{8}-\frac{1}{5}a^{5}-\frac{1}{2}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{3}{10} a^{15} + \frac{4}{5} a^{9} + \frac{39}{10} a^{3} \) (order $4$) | 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: | $\frac{3}{10}a^{16}+\frac{4}{5}a^{10}+\frac{39}{10}a^{4}$, $\frac{1}{5}a^{16}+\frac{1}{10}a^{13}+\frac{7}{10}a^{10}+\frac{1}{10}a^{7}+\frac{31}{10}a^{4}+\frac{9}{5}a$, $\frac{1}{5}a^{17}-\frac{3}{10}a^{16}+\frac{1}{5}a^{14}-\frac{1}{10}a^{12}+\frac{7}{10}a^{11}-\frac{4}{5}a^{10}+\frac{7}{10}a^{8}-\frac{3}{5}a^{6}+\frac{31}{10}a^{5}-\frac{39}{10}a^{4}+\frac{31}{10}a^{2}+a-\frac{13}{10}$, $\frac{3}{10}a^{17}+\frac{1}{5}a^{16}-\frac{1}{5}a^{15}-\frac{1}{10}a^{12}+\frac{4}{5}a^{11}+\frac{7}{10}a^{10}-\frac{7}{10}a^{9}+\frac{1}{2}a^{7}-\frac{1}{10}a^{6}+\frac{39}{10}a^{5}+\frac{31}{10}a^{4}-\frac{31}{10}a^{3}-a^{2}+\frac{1}{2}a+\frac{1}{5}$, $\frac{4}{5}a^{17}-\frac{3}{10}a^{16}-\frac{1}{10}a^{15}+\frac{1}{10}a^{14}-\frac{1}{10}a^{12}+\frac{23}{10}a^{11}-\frac{4}{5}a^{10}-\frac{1}{10}a^{9}+\frac{1}{10}a^{8}-\frac{1}{10}a^{6}+\frac{119}{10}a^{5}-\frac{39}{10}a^{4}-\frac{9}{5}a^{3}+\frac{4}{5}a^{2}+a-\frac{4}{5}$, $\frac{3}{10}a^{17}+\frac{3}{10}a^{16}-\frac{1}{10}a^{15}+\frac{1}{10}a^{12}+\frac{4}{5}a^{11}+\frac{4}{5}a^{10}-\frac{1}{10}a^{9}+\frac{1}{10}a^{6}+\frac{39}{10}a^{5}+\frac{49}{10}a^{4}-\frac{4}{5}a^{3}-a^{2}+\frac{4}{5}$, $\frac{1}{5}a^{17}+\frac{3}{5}a^{16}+\frac{7}{10}a^{15}+\frac{1}{2}a^{14}+\frac{3}{10}a^{13}+\frac{7}{10}a^{11}+\frac{8}{5}a^{10}+\frac{11}{5}a^{9}+\frac{3}{2}a^{8}+\frac{4}{5}a^{7}+\frac{31}{10}a^{5}+\frac{44}{5}a^{4}+\frac{101}{10}a^{3}+8a^{2}+\frac{39}{10}a+1$, $\frac{13}{10}a^{17}+\frac{3}{5}a^{15}+\frac{19}{5}a^{11}+\frac{8}{5}a^{9}+\frac{199}{10}a^{5}+\frac{39}{5}a^{3}$ (assuming GRH) | 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: | \( 428253.6181907014 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 428253.6181907014 \cdot 1}{4\cdot\sqrt{2518170116818978404827136}}\cr\approx \mathstrut & 1.02971569384995 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 36 |
The 12 conjugacy class representatives for $C_6:S_3$ |
Character table for $C_6:S_3$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 3.1.972.1, 3.1.108.1, 3.1.972.2, 3.1.243.1, 6.0.15116544.4, 6.0.15116544.3, 6.0.186624.1, 6.0.3779136.2, 9.1.24794911296.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 18 sibling: | 18.2.7554510350456935214481408.2 |
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 | R | R | ${\href{/padicField/5.2.0.1}{2} }^{8}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.8.1 | $x^{6} + 2 x^{3} + 2$ | $6$ | $1$ | $8$ | $D_{6}$ | $[2]_{3}^{2}$ |
2.12.16.13 | $x^{12} + 10 x^{11} + 47 x^{10} + 144 x^{9} + 330 x^{8} + 578 x^{7} + 785 x^{6} + 830 x^{5} + 530 x^{4} - 64 x^{3} - 189 x^{2} - 30 x + 25$ | $6$ | $2$ | $16$ | $D_6$ | $[2]_{3}^{2}$ | |
\(3\) | Deg $18$ | $9$ | $2$ | $36$ |