Normalized defining polynomial
\( x^{18} - 9 x^{17} + 35 x^{16} - 76 x^{15} + 100 x^{14} - 84 x^{13} + 57 x^{12} - 56 x^{11} + 62 x^{10} + \cdots + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-5460912335827351107\) \(\medspace = -\,3^{9}\cdot 23^{6}\cdot 37^{4}\) | 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: | \(10.99\) | 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: | $3^{1/2}23^{1/2}37^{2/3}\approx 92.23427989230532$ | ||
Ramified primes: | \(3\), \(23\), \(37\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{45}a^{16}+\frac{7}{45}a^{15}-\frac{1}{45}a^{14}-\frac{1}{15}a^{13}+\frac{4}{9}a^{12}+\frac{1}{9}a^{11}+\frac{4}{15}a^{10}-\frac{19}{45}a^{9}+\frac{7}{45}a^{8}+\frac{1}{5}a^{7}+\frac{11}{45}a^{6}-\frac{14}{45}a^{5}+\frac{11}{45}a^{4}+\frac{11}{45}a^{3}-\frac{4}{15}a^{2}+\frac{1}{3}a+\frac{7}{45}$, $\frac{1}{45}a^{17}-\frac{1}{9}a^{15}+\frac{4}{45}a^{14}-\frac{4}{45}a^{13}+\frac{22}{45}a^{11}-\frac{13}{45}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{7}{45}a^{7}-\frac{1}{45}a^{6}+\frac{19}{45}a^{5}-\frac{7}{15}a^{4}+\frac{1}{45}a^{3}+\frac{1}{5}a^{2}-\frac{8}{45}a-\frac{4}{45}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{13946}{45} a^{17} - \frac{118541}{45} a^{16} + \frac{47652}{5} a^{15} - \frac{169138}{9} a^{14} + \frac{972514}{45} a^{13} - \frac{137306}{9} a^{12} + \frac{452902}{45} a^{11} - \frac{111034}{9} a^{10} + \frac{195848}{15} a^{9} - \frac{195172}{45} a^{8} - \frac{54986}{45} a^{7} - \frac{139324}{15} a^{6} + \frac{223166}{15} a^{5} - \frac{112507}{45} a^{4} - \frac{11062}{9} a^{3} - \frac{18536}{5} a^{2} + \frac{139697}{45} a - \frac{28111}{45} \) (order $6$) | 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: | $\frac{3667}{45}a^{17}-\frac{31532}{45}a^{16}+\frac{115706}{45}a^{15}-\frac{15490}{3}a^{14}+\frac{273713}{45}a^{13}-\frac{39713}{9}a^{12}+\frac{43213}{15}a^{11}-\frac{30701}{9}a^{10}+\frac{165973}{45}a^{9}-\frac{20678}{15}a^{8}-\frac{13732}{45}a^{7}-\frac{108134}{45}a^{6}+\frac{187706}{45}a^{5}-\frac{42409}{45}a^{4}-\frac{1124}{3}a^{3}-\frac{4812}{5}a^{2}+\frac{41434}{45}a-\frac{3049}{15}$, $\frac{10459}{45}a^{17}-\frac{88792}{45}a^{16}+\frac{106912}{15}a^{15}-\frac{631132}{45}a^{14}+\frac{144730}{9}a^{13}-\frac{101812}{9}a^{12}+\frac{336293}{45}a^{11}-\frac{414121}{45}a^{10}+\frac{48567}{5}a^{9}-\frac{142874}{45}a^{8}-\frac{41611}{45}a^{7}-\frac{34904}{5}a^{6}+\frac{55396}{5}a^{5}-\frac{80006}{45}a^{4}-\frac{40933}{45}a^{3}-\frac{8372}{3}a^{2}+\frac{103213}{45}a-\frac{4096}{9}$, $\frac{8129}{45}a^{17}-\frac{68339}{45}a^{16}+\frac{81304}{15}a^{15}-\frac{94519}{9}a^{14}+\frac{530806}{45}a^{13}-\frac{72971}{9}a^{12}+\frac{243883}{45}a^{11}-\frac{61768}{9}a^{10}+\frac{106517}{15}a^{9}-\frac{92998}{45}a^{8}-\frac{32579}{45}a^{7}-\frac{27432}{5}a^{6}+\frac{121999}{15}a^{5}-\frac{40978}{45}a^{4}-\frac{5281}{9}a^{3}-\frac{32497}{15}a^{2}+\frac{72263}{45}a-\frac{13249}{45}$, $\frac{2944}{15}a^{17}-\frac{8343}{5}a^{16}+\frac{90587}{15}a^{15}-11917a^{14}+\frac{205826}{15}a^{13}-\frac{29122}{3}a^{12}+\frac{96103}{15}a^{11}-7831a^{10}+\frac{41457}{5}a^{9}-\frac{41648}{15}a^{8}-\frac{11399}{15}a^{7}-\frac{88138}{15}a^{6}+\frac{141502}{15}a^{5}-\frac{24308}{15}a^{4}-\frac{2305}{3}a^{3}-\frac{35036}{15}a^{2}+\frac{29648}{15}a-\frac{6064}{15}$, $\frac{2944}{15}a^{17}-\frac{25019}{15}a^{16}+\frac{30169}{5}a^{15}-\frac{35698}{3}a^{14}+\frac{68457}{5}a^{13}-9680a^{12}+\frac{95918}{15}a^{11}-7822a^{10}+\frac{124081}{15}a^{9}-\frac{41363}{15}a^{8}-\frac{11404}{15}a^{7}-\frac{88183}{15}a^{6}+\frac{141152}{15}a^{5}-\frac{7991}{5}a^{4}-\frac{2282}{3}a^{3}-\frac{35066}{15}a^{2}+\frac{29533}{15}a-\frac{6004}{15}$, $\frac{244}{5}a^{17}-\frac{18893}{45}a^{16}+\frac{69364}{45}a^{15}-\frac{139363}{45}a^{14}+3650a^{13}-\frac{23837}{9}a^{12}+\frac{77792}{45}a^{11}-\frac{10236}{5}a^{10}+\frac{99662}{45}a^{9}-\frac{37301}{45}a^{8}-\frac{2758}{15}a^{7}-\frac{64819}{45}a^{6}+\frac{112771}{45}a^{5}-\frac{25549}{45}a^{4}-\frac{10222}{45}a^{3}-\frac{1738}{3}a^{2}+\frac{2768}{5}a-\frac{1096}{9}$, $\frac{1201}{45}a^{17}-\frac{9608}{45}a^{16}+\frac{32129}{45}a^{15}-\frac{6307}{5}a^{14}+\frac{11074}{9}a^{13}-\frac{6293}{9}a^{12}+\frac{7784}{15}a^{11}-\frac{36229}{45}a^{10}+\frac{32512}{45}a^{9}-\frac{59}{5}a^{8}-\frac{4984}{45}a^{7}-\frac{38549}{45}a^{6}+\frac{38396}{45}a^{5}+\frac{9581}{45}a^{4}+\frac{7}{5}a^{3}-\frac{965}{3}a^{2}+\frac{4777}{45}a$, $8a^{17}-\frac{3796}{45}a^{16}+\frac{17003}{45}a^{15}-\frac{41849}{45}a^{14}+\frac{20296}{15}a^{13}-\frac{10636}{9}a^{12}+\frac{6428}{9}a^{11}-\frac{9704}{15}a^{10}+\frac{37939}{45}a^{9}-\frac{26002}{45}a^{8}-\frac{143}{15}a^{7}-\frac{7376}{45}a^{6}+\frac{40859}{45}a^{5}-\frac{27836}{45}a^{4}-\frac{6656}{45}a^{3}-\frac{402}{5}a^{2}+284a-\frac{4027}{45}$ | 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: | \( 162.382523466 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 162.382523466 \cdot 1}{6\cdot\sqrt{5460912335827351107}}\cr\approx \mathstrut & 0.176756034063 \end{aligned}\]
Galois group
$C_3^3:D_6$ (as 18T137):
A solvable group of order 324 |
The 17 conjugacy class representatives for $C_3^3:D_6$ |
Character table for $C_3^3:D_6$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.1.23.1, 6.0.14283.1, 9.1.449728821.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 9.1.449728821.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(23\) | 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(37\) | 37.3.2.3 | $x^{3} + 111$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
37.3.2.3 | $x^{3} + 111$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
37.6.0.1 | $x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
37.6.0.1 | $x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |