Normalized defining polynomial
\( x^{18} + 27 x^{16} + 300 x^{14} + 1790 x^{12} + 6291 x^{10} + 13431 x^{8} + 17229 x^{6} + 12465 x^{4} + \cdots + 375 \)
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: |
\(-374995864672776683690097475584000\)
\(\medspace = -\,2^{18}\cdot 3^{27}\cdot 5^{3}\cdot 107^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(64.52\) | 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: | not computed | ||
Ramified primes: |
\(2\), \(3\), \(5\), \(107\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-15}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{256}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{10}a^{14}+\frac{1}{5}a^{12}-\frac{2}{5}a^{6}-\frac{2}{5}a^{4}+\frac{2}{5}a^{2}-\frac{1}{2}$, $\frac{1}{10}a^{15}+\frac{1}{5}a^{13}-\frac{2}{5}a^{7}-\frac{2}{5}a^{5}+\frac{2}{5}a^{3}-\frac{1}{2}a$, $\frac{1}{100}a^{16}+\frac{1}{50}a^{14}-\frac{1}{10}a^{10}-\frac{9}{100}a^{8}-\frac{11}{25}a^{6}-\frac{21}{100}a^{4}+\frac{2}{5}a^{2}+\frac{1}{4}$, $\frac{1}{100}a^{17}+\frac{1}{50}a^{15}-\frac{1}{10}a^{11}-\frac{9}{100}a^{9}-\frac{11}{25}a^{7}-\frac{21}{100}a^{5}+\frac{2}{5}a^{3}+\frac{1}{4}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{308}$, which has order $2464$ (assuming GRH)
Unit group
Rank: | $8$ | 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: |
$\frac{53}{25}a^{16}+\frac{1346}{25}a^{14}+\frac{5497}{10}a^{12}+\frac{29143}{10}a^{10}+\frac{216823}{25}a^{8}+\frac{365208}{25}a^{6}+\frac{659929}{50}a^{4}+\frac{26871}{5}a^{2}+\frac{995}{2}$, $\frac{111}{100}a^{16}+\frac{1411}{50}a^{14}+\frac{577}{2}a^{12}+\frac{15319}{10}a^{10}+\frac{456701}{100}a^{8}+\frac{385383}{50}a^{6}+\frac{698069}{100}a^{4}+\frac{28539}{10}a^{2}+\frac{1073}{4}$, $\frac{243}{100}a^{16}+\frac{1539}{25}a^{14}+\frac{3132}{5}a^{12}+\frac{33057}{10}a^{10}+\frac{977763}{100}a^{8}+\frac{408532}{25}a^{6}+\frac{1463067}{100}a^{4}+\frac{11805}{2}a^{2}+\frac{2179}{4}$, $\frac{1}{5}a^{16}+\frac{24}{5}a^{14}+\frac{453}{10}a^{12}+215a^{10}+\frac{5477}{10}a^{8}+\frac{7441}{10}a^{6}+\frac{5037}{10}a^{4}+\frac{703}{5}a^{2}+8$, $\frac{37}{50}a^{16}+\frac{482}{25}a^{14}+\frac{1018}{5}a^{12}+\frac{11281}{10}a^{10}+\frac{177567}{50}a^{8}+\frac{320287}{50}a^{6}+\frac{156469}{25}a^{4}+\frac{27623}{10}a^{2}+\frac{521}{2}$, $\frac{243}{100}a^{16}+\frac{3063}{50}a^{14}+\frac{6193}{10}a^{12}+\frac{16201}{5}a^{10}+\frac{947863}{100}a^{8}+\frac{390787}{25}a^{6}+\frac{1379037}{100}a^{4}+\frac{54853}{10}a^{2}+\frac{2023}{4}$, $\frac{121}{100}a^{16}+\frac{768}{25}a^{14}+\frac{1568}{5}a^{12}+\frac{16629}{10}a^{10}+\frac{495261}{100}a^{8}+\frac{208864}{25}a^{6}+\frac{756289}{100}a^{4}+\frac{30851}{10}a^{2}+\frac{1145}{4}$, $\frac{133}{50}a^{16}+\frac{3391}{50}a^{14}+696a^{12}+\frac{18572}{5}a^{10}+\frac{278614}{25}a^{8}+\frac{473674}{25}a^{6}+\frac{431941}{25}a^{4}+\frac{70819}{10}a^{2}+\frac{1297}{2}$
| 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: | \( 664139.073753 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 664139.073753 \cdot 2464}{2\cdot\sqrt{374995864672776683690097475584000}}\cr\approx \mathstrut & 0.644874665987 \end{aligned}\] (assuming GRH)
Galois group
$C_2^4:(C_6\times S_4)$ (as 18T367):
A solvable group of order 2304 |
The 48 conjugacy class representatives for $C_2^4:(C_6\times S_4)$ |
Character table for $C_2^4:(C_6\times S_4)$ |
Intermediate fields
\(\Q(\zeta_{9})^+\), 3.3.321.1, 9.9.1953114230889.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Degree 36 siblings: | data not computed |
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 | R | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.6.6.4 | $x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
2.6.6.2 | $x^{6} + 6 x^{5} + 14 x^{4} + 24 x^{3} + 44 x^{2} + 8 x + 72$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2]^{6}$ | |
2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
\(3\)
| 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.12.19.43 | $x^{12} + 6 x^{8} + 3$ | $12$ | $1$ | $19$ | $D_4 \times C_3$ | $[2]_{4}^{2}$ | |
\(5\)
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(107\)
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
107.4.2.1 | $x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
107.4.2.1 | $x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
107.4.2.1 | $x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |