Normalized defining polynomial
\( x^{18} - 3 x^{17} + 12 x^{16} - 21 x^{15} + 51 x^{14} - 60 x^{13} + 114 x^{12} - 75 x^{11} + 165 x^{10} + \cdots + 1 \)
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: | \(-239338060927275176307\) \(\medspace = -\,3^{25}\cdot 7^{10}\) | 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: | \(13.56\) | 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: | $3^{25/18}7^{2/3}\approx 16.82934070487988$ | ||
Ramified primes: | \(3\), \(7\) | 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: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{1600394}a^{17}+\frac{41817}{800197}a^{16}+\frac{374893}{1600394}a^{15}+\frac{3286}{800197}a^{14}+\frac{727273}{1600394}a^{13}+\frac{757083}{1600394}a^{12}-\frac{118911}{800197}a^{11}-\frac{110930}{800197}a^{10}+\frac{31789}{800197}a^{9}+\frac{164117}{1600394}a^{8}+\frac{237281}{800197}a^{7}-\frac{214752}{800197}a^{6}-\frac{391193}{800197}a^{5}+\frac{246049}{800197}a^{4}-\frac{532197}{1600394}a^{3}+\frac{397881}{1600394}a^{2}+\frac{290388}{800197}a+\frac{3827}{1600394}$
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: | \( -\frac{289}{2378} a^{17} + \frac{955}{2378} a^{16} - \frac{604}{1189} a^{15} - \frac{832}{1189} a^{14} + \frac{14279}{2378} a^{13} - \frac{17033}{1189} a^{12} + \frac{37660}{1189} a^{11} - \frac{51364}{1189} a^{10} + \frac{75289}{1189} a^{9} - \frac{112369}{2378} a^{8} + \frac{172759}{2378} a^{7} - \frac{10889}{2378} a^{6} + \frac{129403}{2378} a^{5} + \frac{65363}{2378} a^{4} + \frac{33022}{1189} a^{3} + \frac{43385}{2378} a^{2} + \frac{19945}{2378} a + \frac{2856}{1189} \) (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: | $\frac{1086095}{1600394}a^{17}-\frac{1146908}{800197}a^{16}+\frac{4587155}{800197}a^{15}-\frac{4771632}{800197}a^{14}+\frac{29114175}{1600394}a^{13}-\frac{15175527}{1600394}a^{12}+\frac{59919709}{1600394}a^{11}+\frac{3086107}{1600394}a^{10}+\frac{118777345}{1600394}a^{9}+\frac{32993464}{800197}a^{8}+\frac{181369257}{1600394}a^{7}+\frac{62763486}{800197}a^{6}+\frac{162115561}{1600394}a^{5}+\frac{55612522}{800197}a^{4}+\frac{39310081}{800197}a^{3}+\frac{44588241}{1600394}a^{2}+\frac{8111447}{800197}a+\frac{1331469}{800197}$, $\frac{792107}{800197}a^{17}-\frac{4866175}{1600394}a^{16}+\frac{9464624}{800197}a^{15}-\frac{17158615}{800197}a^{14}+\frac{41020018}{800197}a^{13}-\frac{105013071}{1600394}a^{12}+\frac{191859867}{1600394}a^{11}-\frac{154426963}{1600394}a^{10}+\frac{286031631}{1600394}a^{9}-\frac{77978523}{1600394}a^{8}+\frac{139372904}{800197}a^{7}+\frac{54077171}{1600394}a^{6}+\frac{85565549}{800197}a^{5}+\frac{87035983}{1600394}a^{4}+\frac{31621553}{800197}a^{3}+\frac{17939575}{800197}a^{2}+\frac{9360055}{1600394}a+\frac{1047450}{800197}$, $\frac{44}{1189}a^{17}+\frac{1130}{1189}a^{16}-\frac{7733}{2378}a^{15}+\frac{14509}{1189}a^{14}-\frac{25703}{1189}a^{13}+\frac{54133}{1189}a^{12}-\frac{124403}{2378}a^{11}+\frac{198263}{2378}a^{10}-\frac{113525}{2378}a^{9}+\frac{223045}{2378}a^{8}+\frac{16855}{2378}a^{7}+\frac{84209}{1189}a^{6}+\frac{96575}{2378}a^{5}+\frac{43426}{1189}a^{4}+\frac{57257}{2378}a^{3}+\frac{13007}{1189}a^{2}+\frac{3723}{1189}a+\frac{289}{2378}$, $\frac{469304}{800197}a^{17}-\frac{1492508}{800197}a^{16}+\frac{5071461}{800197}a^{15}-\frac{7695320}{800197}a^{14}+\frac{14504143}{800197}a^{13}-\frac{10193875}{800197}a^{12}+\frac{14265021}{800197}a^{11}+\frac{13050958}{800197}a^{10}+\frac{1264370}{800197}a^{9}+\frac{48214744}{800197}a^{8}-\frac{1785374}{800197}a^{7}+\frac{49690904}{800197}a^{6}+\frac{8717849}{800197}a^{5}+\frac{22709532}{800197}a^{4}+\frac{10510495}{800197}a^{3}+\frac{4375662}{800197}a^{2}+\frac{2198946}{800197}a+\frac{1184537}{800197}$, $\frac{34061}{1600394}a^{17}-\frac{822020}{800197}a^{16}+\frac{6088323}{1600394}a^{15}-\frac{10505695}{800197}a^{14}+\frac{40757171}{1600394}a^{13}-\frac{81764553}{1600394}a^{12}+\frac{52382448}{800197}a^{11}-\frac{76675408}{800197}a^{10}+\frac{57712772}{800197}a^{9}-\frac{163427293}{1600394}a^{8}+\frac{13641790}{800197}a^{7}-\frac{51279703}{800197}a^{6}-\frac{25150633}{800197}a^{5}-\frac{19793117}{800197}a^{4}-\frac{42709817}{1600394}a^{3}-\frac{11114409}{1600394}a^{2}-\frac{4330434}{800197}a-\frac{880861}{1600394}$, $\frac{1187783}{1600394}a^{17}-\frac{1106670}{800197}a^{16}+\frac{4954307}{800197}a^{15}-\frac{5107210}{800197}a^{14}+\frac{36245835}{1600394}a^{13}-\frac{22274369}{1600394}a^{12}+\frac{82235423}{1600394}a^{11}-\frac{4661325}{1600394}a^{10}+\frac{151913523}{1600394}a^{9}+\frac{47307442}{800197}a^{8}+\frac{212357117}{1600394}a^{7}+\frac{95246317}{800197}a^{6}+\frac{182439643}{1600394}a^{5}+\frac{78489151}{800197}a^{4}+\frac{43748550}{800197}a^{3}+\frac{54353019}{1600394}a^{2}+\frac{8216697}{800197}a+\frac{663389}{800197}$, $\frac{25659}{55186}a^{17}-\frac{80769}{55186}a^{16}+\frac{160861}{27593}a^{15}-\frac{284664}{27593}a^{14}+\frac{1331657}{55186}a^{13}-\frac{746796}{27593}a^{12}+\frac{1365869}{27593}a^{11}-\frac{741966}{27593}a^{10}+\frac{1790823}{27593}a^{9}+\frac{717419}{55186}a^{8}+\frac{3886071}{55186}a^{7}+\frac{3174259}{55186}a^{6}+\frac{3604273}{55186}a^{5}+\frac{3083047}{55186}a^{4}+\frac{1128769}{27593}a^{3}+\frac{1308601}{55186}a^{2}+\frac{669699}{55186}a+\frac{79532}{27593}$, $\frac{743375}{1600394}a^{17}-\frac{1481559}{1600394}a^{16}+\frac{3538032}{800197}a^{15}-\frac{4272176}{800197}a^{14}+\frac{28274357}{1600394}a^{13}-\frac{9792117}{800197}a^{12}+\frac{31105057}{800197}a^{11}-\frac{1487703}{800197}a^{10}+\frac{50942679}{800197}a^{9}+\frac{85660743}{1600394}a^{8}+\frac{144312199}{1600394}a^{7}+\frac{179312113}{1600394}a^{6}+\frac{147428811}{1600394}a^{5}+\frac{156129827}{1600394}a^{4}+\frac{45337684}{800197}a^{3}+\frac{60386631}{1600394}a^{2}+\frac{25676909}{1600394}a+\frac{2498486}{800197}$ | 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: | \( 2026.39351046 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 2026.39351046 \cdot 1}{6\cdot\sqrt{239338060927275176307}}\cr\approx \mathstrut & 0.333185019566 \end{aligned}\]
Galois group
$C_3^2:C_6$ (as 18T22):
A solvable group of order 54 |
The 10 conjugacy class representatives for $C_3^2:C_6$ |
Character table for $C_3^2:C_6$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 6.0.107163.1 |
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 sibling: | data not computed |
Minimal sibling: | 9.1.62523502209.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $18$ | $18$ | $1$ | $25$ | |||
\(7\) | 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |