Normalized defining polynomial
\( x^{18} - 2 x^{17} - 4 x^{16} + 12 x^{15} - 20 x^{14} + 28 x^{13} + 84 x^{12} - 232 x^{11} - 180 x^{10} + 900 x^{9} - 334 x^{8} - 1660 x^{7} + 2996 x^{6} - 2576 x^{5} + 1380 x^{4} + \cdots + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(4581441688047722385682249\) \(\medspace = 7^{16}\cdot 13^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.45\) | 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: | $7^{8/9}13^{5/6}\approx 47.806217686235044$ | ||
Ramified primes: | \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{6}a+\frac{1}{6}$, $\frac{1}{6}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{6}a^{2}-\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{6}a^{11}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}+\frac{1}{3}$, $\frac{1}{6}a^{12}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}$, $\frac{1}{6}a^{13}-\frac{1}{3}a^{7}-\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{3}$, $\frac{1}{42}a^{14}-\frac{1}{14}a^{13}+\frac{1}{14}a^{12}+\frac{1}{21}a^{10}-\frac{1}{42}a^{9}+\frac{5}{21}a^{8}+\frac{2}{7}a^{7}-\frac{19}{42}a^{6}-\frac{1}{21}a^{5}+\frac{5}{14}a^{3}-\frac{2}{21}a^{2}+\frac{5}{42}a-\frac{17}{42}$, $\frac{1}{42}a^{15}+\frac{1}{42}a^{13}+\frac{1}{21}a^{12}+\frac{1}{21}a^{11}-\frac{1}{21}a^{10}+\frac{1}{3}a^{8}-\frac{11}{42}a^{7}-\frac{17}{42}a^{6}-\frac{13}{42}a^{5}+\frac{1}{42}a^{4}+\frac{1}{7}a^{3}-\frac{1}{3}a^{2}-\frac{1}{21}a-\frac{8}{21}$, $\frac{1}{42}a^{16}-\frac{1}{21}a^{13}-\frac{1}{42}a^{12}-\frac{1}{21}a^{11}-\frac{1}{21}a^{10}+\frac{1}{42}a^{9}-\frac{1}{6}a^{8}-\frac{1}{42}a^{7}+\frac{10}{21}a^{6}-\frac{3}{7}a^{5}+\frac{13}{42}a^{4}+\frac{13}{42}a^{3}+\frac{1}{21}a^{2}+\frac{1}{6}a-\frac{11}{42}$, $\frac{1}{42}a^{17}-\frac{1}{14}a^{12}-\frac{1}{21}a^{11}-\frac{1}{21}a^{10}-\frac{1}{21}a^{9}+\frac{19}{42}a^{8}+\frac{1}{21}a^{7}+\frac{1}{3}a^{6}-\frac{2}{7}a^{5}-\frac{1}{42}a^{4}-\frac{1}{14}a^{3}-\frac{4}{21}a^{2}+\frac{13}{42}a+\frac{5}{14}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $11$ | 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: | $\frac{188}{21}a^{17}-\frac{274}{21}a^{16}-\frac{1825}{42}a^{15}+\frac{1775}{21}a^{14}-\frac{913}{7}a^{13}+\frac{2455}{14}a^{12}+\frac{17954}{21}a^{11}-\frac{34073}{21}a^{10}-\frac{53476}{21}a^{9}+6762a^{8}+\frac{35527}{42}a^{7}-\frac{620531}{42}a^{6}+\frac{392870}{21}a^{5}-\frac{506395}{42}a^{4}+\frac{28703}{6}a^{3}-\frac{24398}{21}a^{2}+\frac{3134}{21}a-\frac{130}{21}$, $\frac{188}{21}a^{17}-\frac{274}{21}a^{16}-\frac{1825}{42}a^{15}+\frac{1775}{21}a^{14}-\frac{913}{7}a^{13}+\frac{2455}{14}a^{12}+\frac{17954}{21}a^{11}-\frac{34073}{21}a^{10}-\frac{53476}{21}a^{9}+6762a^{8}+\frac{35527}{42}a^{7}-\frac{620531}{42}a^{6}+\frac{392870}{21}a^{5}-\frac{506395}{42}a^{4}+\frac{28703}{6}a^{3}-\frac{24398}{21}a^{2}+\frac{3134}{21}a-\frac{109}{21}$, $\frac{1271}{21}a^{17}-\frac{635}{6}a^{16}-\frac{11299}{42}a^{15}+\frac{13840}{21}a^{14}-\frac{6259}{6}a^{13}+\frac{10009}{7}a^{12}+\frac{228805}{42}a^{11}-\frac{88743}{7}a^{10}-\frac{592757}{42}a^{9}+\frac{2140877}{42}a^{8}-\frac{153313}{21}a^{7}-\frac{615107}{6}a^{6}+\frac{2176845}{14}a^{5}-\frac{1628595}{14}a^{4}+\frac{322259}{6}a^{3}-\frac{223051}{14}a^{2}+\frac{118745}{42}a-\frac{3193}{14}$, $\frac{361}{6}a^{17}-\frac{2062}{21}a^{16}-\frac{1943}{7}a^{15}+\frac{4343}{7}a^{14}-\frac{6798}{7}a^{13}+\frac{3968}{3}a^{12}+\frac{77701}{14}a^{11}-\frac{500855}{42}a^{10}-\frac{107067}{7}a^{9}+\frac{2041457}{42}a^{8}-\frac{41098}{21}a^{7}-\frac{2120921}{21}a^{6}+142830a^{5}-\frac{709798}{7}a^{4}+\frac{625637}{14}a^{3}-\frac{264532}{21}a^{2}+\frac{43847}{21}a-\frac{2159}{14}$, $a-1$, $a^{17}-a^{16}-5a^{15}+7a^{14}-13a^{13}+15a^{12}+99a^{11}-133a^{10}-313a^{9}+587a^{8}+253a^{7}-1407a^{6}+1589a^{5}-987a^{4}+393a^{3}-99a^{2}+16a-1$, $\frac{341}{14}a^{17}-\frac{275}{6}a^{16}-\frac{4369}{42}a^{15}+\frac{3937}{14}a^{14}-\frac{2689}{6}a^{13}+\frac{13024}{21}a^{12}+\frac{44840}{21}a^{11}-\frac{227461}{42}a^{10}-\frac{107914}{21}a^{9}+\frac{451054}{21}a^{8}-\frac{110095}{21}a^{7}-41784a^{6}+\frac{474512}{7}a^{5}-\frac{742017}{14}a^{4}+\frac{76043}{3}a^{3}-\frac{326909}{42}a^{2}+\frac{30200}{21}a-\frac{5071}{42}$, $\frac{341}{14}a^{17}-\frac{275}{6}a^{16}-\frac{4369}{42}a^{15}+\frac{3937}{14}a^{14}-\frac{2689}{6}a^{13}+\frac{13024}{21}a^{12}+\frac{44840}{21}a^{11}-\frac{227461}{42}a^{10}-\frac{107914}{21}a^{9}+\frac{451054}{21}a^{8}-\frac{110095}{21}a^{7}-41784a^{6}+\frac{474512}{7}a^{5}-\frac{742017}{14}a^{4}+\frac{76043}{3}a^{3}-\frac{326909}{42}a^{2}+\frac{30200}{21}a-\frac{5113}{42}$, $\frac{1657}{14}a^{17}-\frac{8167}{42}a^{16}-\frac{3803}{7}a^{15}+\frac{17173}{14}a^{14}-\frac{80933}{42}a^{13}+\frac{55088}{21}a^{12}+\frac{228556}{21}a^{11}-\frac{330089}{14}a^{10}-\frac{416719}{14}a^{9}+\frac{2014811}{21}a^{8}-\frac{71829}{14}a^{7}-\frac{8338571}{42}a^{6}+\frac{5954584}{21}a^{5}-\frac{2842677}{14}a^{4}+\frac{1895596}{21}a^{3}-\frac{154291}{6}a^{2}+\frac{182291}{42}a-\frac{6899}{21}$, $\frac{166}{3}a^{17}-\frac{1327}{14}a^{16}-\frac{5210}{21}a^{15}+\frac{12448}{21}a^{14}-\frac{19715}{21}a^{13}+\frac{26902}{21}a^{12}+\frac{70157}{14}a^{11}-\frac{239345}{21}a^{10}-\frac{553793}{42}a^{9}+\frac{1931879}{42}a^{8}-\frac{228425}{42}a^{7}-\frac{652992}{7}a^{6}+\frac{2924261}{21}a^{5}-\frac{2163598}{21}a^{4}+\frac{1976939}{42}a^{3}-\frac{578695}{42}a^{2}+\frac{4807}{2}a-\frac{7999}{42}$, $\frac{1879}{21}a^{17}-\frac{1070}{7}a^{16}-\frac{5623}{14}a^{15}+\frac{20078}{21}a^{14}-\frac{63625}{42}a^{13}+\frac{43423}{21}a^{12}+\frac{340577}{42}a^{11}-\frac{772181}{42}a^{10}-\frac{898103}{42}a^{9}+\frac{222646}{3}a^{8}-\frac{357671}{42}a^{7}-\frac{2108853}{14}a^{6}+\frac{3142289}{14}a^{5}-\frac{6974741}{42}a^{4}+\frac{456283}{6}a^{3}-\frac{470104}{21}a^{2}+\frac{82865}{21}a-\frac{13367}{42}$ | 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: | \( 299765.553553 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 299765.553553 \cdot 1}{2\cdot\sqrt{4581441688047722385682249}}\cr\approx \mathstrut & 0.275746388337 \end{aligned}\]
Galois group
$C_3^2.A_4$ (as 18T47):
A solvable group of order 108 |
The 20 conjugacy class representatives for $C_3^2.A_4$ |
Character table for $C_3^2.A_4$ |
Intermediate fields
\(\Q(\zeta_{7})^+\), 6.2.405769.1, 9.9.164648481361.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 siblings: | data not computed |
Degree 36 sibling: | 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 | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.9.8.2 | $x^{9} + 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
7.9.8.2 | $x^{9} + 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ | |
\(13\) | 13.6.5.4 | $x^{6} + 26$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
13.6.5.1 | $x^{6} + 52$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |