Normalized defining polynomial
\( x^{18} - 54 x^{16} + 1215 x^{14} - 14742 x^{12} + 104247 x^{10} - 136 x^{9} - 433026 x^{8} + 3672 x^{7} + 1010394 x^{6} - 33048 x^{5} - 1180980 x^{4} + 110160 x^{3} + \cdots - 40553 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(6966114824903498893840251683313\) \(\medspace = 3^{45}\cdot 11^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(51.70\) | 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: | $3^{5/2}11^{1/2}\approx 51.70106381884226$ | ||
Ramified primes: | \(3\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{33}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(297=3^{3}\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{297}(1,·)$, $\chi_{297}(67,·)$, $\chi_{297}(133,·)$, $\chi_{297}(65,·)$, $\chi_{297}(265,·)$, $\chi_{297}(34,·)$, $\chi_{297}(131,·)$, $\chi_{297}(100,·)$, $\chi_{297}(199,·)$, $\chi_{297}(197,·)$, $\chi_{297}(32,·)$, $\chi_{297}(98,·)$, $\chi_{297}(164,·)$, $\chi_{297}(166,·)$, $\chi_{297}(230,·)$, $\chi_{297}(232,·)$, $\chi_{297}(263,·)$, $\chi_{297}(296,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{74}a^{9}-\frac{27}{74}a^{7}+\frac{21}{74}a^{5}+\frac{2}{37}a^{3}-\frac{11}{74}a-\frac{31}{74}$, $\frac{1}{74}a^{10}-\frac{27}{74}a^{8}+\frac{21}{74}a^{6}+\frac{2}{37}a^{4}-\frac{11}{74}a^{2}-\frac{31}{74}a$, $\frac{1}{74}a^{11}+\frac{16}{37}a^{7}-\frac{21}{74}a^{5}+\frac{23}{74}a^{3}-\frac{31}{74}a^{2}-\frac{1}{74}a-\frac{23}{74}$, $\frac{1}{74}a^{12}+\frac{16}{37}a^{8}-\frac{21}{74}a^{6}+\frac{23}{74}a^{4}-\frac{31}{74}a^{3}-\frac{1}{74}a^{2}-\frac{23}{74}a$, $\frac{1}{74}a^{13}+\frac{29}{74}a^{7}+\frac{17}{74}a^{5}-\frac{31}{74}a^{4}+\frac{19}{74}a^{3}-\frac{23}{74}a^{2}-\frac{9}{37}a+\frac{15}{37}$, $\frac{1}{87838}a^{14}-\frac{206}{43919}a^{13}-\frac{21}{43919}a^{12}-\frac{275}{43919}a^{11}-\frac{247}{43919}a^{10}-\frac{59}{87838}a^{9}+\frac{26379}{87838}a^{8}-\frac{32099}{87838}a^{7}-\frac{1113}{87838}a^{6}-\frac{15937}{43919}a^{5}+\frac{31901}{87838}a^{4}+\frac{26445}{87838}a^{3}-\frac{4099}{43919}a^{2}+\frac{9015}{87838}a-\frac{41171}{87838}$, $\frac{1}{87838}a^{15}-\frac{45}{87838}a^{13}-\frac{49}{87838}a^{12}-\frac{377}{87838}a^{11}+\frac{577}{87838}a^{10}-\frac{303}{87838}a^{9}+\frac{8235}{87838}a^{8}-\frac{9076}{43919}a^{7}-\frac{2573}{87838}a^{6}-\frac{5203}{87838}a^{5}-\frac{35}{2374}a^{4}-\frac{32071}{87838}a^{3}+\frac{18471}{43919}a^{2}+\frac{2805}{87838}a+\frac{29453}{87838}$, $\frac{1}{87838}a^{16}+\frac{403}{87838}a^{13}+\frac{107}{87838}a^{12}-\frac{433}{87838}a^{11}+\frac{10}{43919}a^{10}-\frac{355}{87838}a^{9}+\frac{10537}{43919}a^{8}+\frac{23665}{87838}a^{7}+\frac{17119}{87838}a^{6}+\frac{10412}{43919}a^{5}-\frac{3341}{43919}a^{4}+\frac{16227}{87838}a^{3}-\frac{16279}{43919}a^{2}-\frac{2031}{43919}a+\frac{11976}{43919}$, $\frac{1}{87838}a^{17}-\frac{1}{2374}a^{13}-\frac{125}{87838}a^{12}-\frac{299}{87838}a^{11}+\frac{249}{43919}a^{10}-\frac{255}{87838}a^{9}+\frac{10643}{87838}a^{8}-\frac{17569}{43919}a^{7}-\frac{3312}{43919}a^{6}+\frac{24875}{87838}a^{5}+\frac{4106}{43919}a^{4}+\frac{1895}{43919}a^{3}+\frac{18928}{43919}a^{2}+\frac{39745}{87838}a+\frac{7729}{43919}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $17$ | 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: | $\frac{136}{43919}a^{15}-\frac{6120}{43919}a^{13}-\frac{729}{43919}a^{12}+\frac{110160}{43919}a^{11}+\frac{26244}{43919}a^{10}-\frac{1009800}{43919}a^{9}-\frac{354294}{43919}a^{8}+\frac{4957200}{43919}a^{7}+\frac{2205683}{43919}a^{6}-\frac{12492144}{43919}a^{5}-\frac{6221511}{43919}a^{4}+\frac{13870664}{43919}a^{3}+\frac{6473439}{43919}a^{2}-\frac{4376016}{43919}a-\frac{1126980}{43919}$, $\frac{80}{43919}a^{15}-\frac{3600}{43919}a^{13}-\frac{359}{43919}a^{12}+\frac{64800}{43919}a^{11}+\frac{12924}{43919}a^{10}-\frac{594000}{43919}a^{9}-\frac{174474}{43919}a^{8}+\frac{2916000}{43919}a^{7}+\frac{1085616}{43919}a^{6}-\frac{7348320}{43919}a^{5}-\frac{3053295}{43919}a^{4}+\frac{8164800}{43919}a^{3}+\frac{3140532}{43919}a^{2}-\frac{2624400}{43919}a-\frac{523422}{43919}$, $\frac{31}{87838}a^{17}+\frac{117}{87838}a^{16}-\frac{1741}{87838}a^{15}-\frac{5697}{87838}a^{14}+\frac{40537}{87838}a^{13}+\frac{56816}{43919}a^{12}-\frac{253313}{43919}a^{11}-\frac{597875}{43919}a^{10}+\frac{3673689}{87838}a^{9}+\frac{3558534}{43919}a^{8}-\frac{15558173}{87838}a^{7}-\frac{23773113}{87838}a^{6}+\frac{18265951}{43919}a^{5}+\frac{41055443}{87838}a^{4}-\frac{41104533}{87838}a^{3}-\frac{29227421}{87838}a^{2}+\frac{14678451}{87838}a+\frac{1918906}{43919}$, $\frac{31}{87838}a^{17}+\frac{253}{87838}a^{16}-\frac{1149}{87838}a^{15}-\frac{12063}{87838}a^{14}+\frac{13625}{87838}a^{13}+\frac{115615}{43919}a^{12}-\frac{505}{2374}a^{11}-\frac{1135922}{43919}a^{10}-\frac{404430}{43919}a^{9}+\frac{6031302}{43919}a^{8}+\frac{85149}{1187}a^{7}-\frac{901883}{2374}a^{6}-\frac{8660887}{43919}a^{5}+\frac{41993515}{87838}a^{4}+\frac{7769579}{43919}a^{3}-\frac{8461671}{43919}a^{2}-\frac{993267}{87838}a+\frac{215425}{43919}$, $\frac{105}{87838}a^{17}-\frac{68}{43919}a^{16}-\frac{4923}{87838}a^{15}+\frac{2924}{43919}a^{14}+\frac{47115}{43919}a^{13}-\frac{50456}{43919}a^{12}-\frac{474174}{43919}a^{11}+\frac{447440}{43919}a^{10}+\frac{5383179}{87838}a^{9}-\frac{4328509}{87838}a^{8}-\frac{463085}{2374}a^{7}+\frac{5534723}{43919}a^{6}+\frac{28355951}{87838}a^{5}-\frac{12834109}{87838}a^{4}-\frac{19959359}{87838}a^{3}+\frac{3791851}{87838}a^{2}+\frac{1647490}{43919}a+\frac{210346}{43919}$, $\frac{117}{87838}a^{16}-\frac{2808}{43919}a^{14}+\frac{54756}{43919}a^{12}-\frac{884}{43919}a^{11}-\frac{555984}{43919}a^{10}+\frac{29172}{43919}a^{9}+\frac{3127410}{43919}a^{8}-\frac{701315}{87838}a^{7}-\frac{9552816}{43919}a^{6}+\frac{3700599}{87838}a^{5}+\frac{14329224}{43919}a^{4}-\frac{4013001}{43919}a^{3}-\frac{16391687}{87838}a^{2}+\frac{4950207}{87838}a+\frac{813930}{43919}$, $\frac{599}{87838}a^{14}-\frac{1079}{87838}a^{13}-\frac{12579}{43919}a^{12}+\frac{42081}{87838}a^{11}+\frac{415107}{87838}a^{10}-\frac{631215}{87838}a^{9}-\frac{1698165}{43919}a^{8}+\frac{2272374}{43919}a^{7}+\frac{7132293}{43919}a^{6}-\frac{7953309}{43919}a^{5}-\frac{14264586}{43919}a^{4}+\frac{23859927}{87838}a^{3}+\frac{21396879}{87838}a^{2}-\frac{10225683}{87838}a-\frac{1310013}{43919}$, $\frac{340}{43919}a^{14}-\frac{1215}{87838}a^{13}-\frac{14280}{43919}a^{12}+\frac{47385}{87838}a^{11}+\frac{235620}{43919}a^{10}-\frac{710775}{87838}a^{9}-\frac{1927800}{43919}a^{8}+\frac{2558790}{43919}a^{7}+\frac{8096760}{43919}a^{6}-\frac{483935}{2374}a^{5}-\frac{32388227}{87838}a^{4}+\frac{13389135}{43919}a^{3}+\frac{12152262}{43919}a^{2}-\frac{5623740}{43919}a-\frac{1497843}{43919}$, $\frac{117}{87838}a^{16}-\frac{759}{87838}a^{15}-\frac{2468}{43919}a^{14}+\frac{16470}{43919}a^{13}+\frac{84907}{87838}a^{12}-\frac{569173}{87838}a^{11}-\frac{391554}{43919}a^{10}+\frac{2491572}{43919}a^{9}+\frac{2160675}{43919}a^{8}-\frac{23249285}{87838}a^{7}-\frac{7436016}{43919}a^{6}+\frac{27756095}{43919}a^{5}+\frac{14953748}{43919}a^{4}-\frac{29355636}{43919}a^{3}-\frac{26685503}{87838}a^{2}+\frac{9300861}{43919}a+\frac{2199282}{43919}$, $\frac{68}{43919}a^{17}-\frac{253}{87838}a^{16}-\frac{5745}{87838}a^{15}+\frac{5084}{43919}a^{14}+\frac{47726}{43919}a^{13}-\frac{159947}{87838}a^{12}-\frac{393304}{43919}a^{11}+\frac{1249299}{87838}a^{10}+\frac{3310005}{87838}a^{9}-\frac{5103109}{87838}a^{8}-\frac{3192309}{43919}a^{7}+\frac{10704739}{87838}a^{6}+\frac{1572661}{43919}a^{5}-\frac{5476801}{43919}a^{4}+\frac{3688537}{87838}a^{3}+\frac{4612533}{87838}a^{2}-\frac{2795113}{87838}a+\frac{43982}{43919}$, $\frac{55}{87838}a^{17}+\frac{155}{87838}a^{16}-\frac{2989}{87838}a^{15}-\frac{7721}{87838}a^{14}+\frac{33909}{43919}a^{13}+\frac{157589}{87838}a^{12}-\frac{832703}{87838}a^{11}-\frac{45811}{2374}a^{10}+\frac{2993964}{43919}a^{9}+\frac{10273045}{87838}a^{8}-\frac{25396941}{87838}a^{7}-\frac{34647037}{87838}a^{6}+\frac{60191875}{87838}a^{5}+\frac{59427871}{87838}a^{4}-\frac{34131640}{43919}a^{3}-\frac{40825129}{87838}a^{2}+\frac{23572155}{87838}a+\frac{2622215}{43919}$, $\frac{31}{87838}a^{17}-\frac{1989}{87838}a^{15}+\frac{340}{43919}a^{14}+\frac{25173}{43919}a^{13}-\frac{26373}{87838}a^{12}-\frac{653049}{87838}a^{11}+\frac{390707}{87838}a^{10}+\frac{2333205}{43919}a^{9}-\frac{1369344}{43919}a^{8}-\frac{9103023}{43919}a^{7}+\frac{4504578}{43919}a^{6}+\frac{35706523}{87838}a^{5}-\frac{5646172}{43919}a^{4}-\frac{28708229}{87838}a^{3}+\frac{618591}{43919}a^{2}+\frac{2866743}{43919}a+\frac{630270}{43919}$, $\frac{1}{1187}a^{17}-\frac{141}{87838}a^{16}-\frac{3535}{87838}a^{15}+\frac{6985}{87838}a^{14}+\frac{34060}{43919}a^{13}-\frac{71059}{43919}a^{12}-\frac{336505}{43919}a^{11}+\frac{762247}{43919}a^{10}+\frac{3550845}{87838}a^{9}-\frac{9177901}{87838}a^{8}-\frac{4589236}{43919}a^{7}+\frac{15167269}{43919}a^{6}+\frac{7327323}{87838}a^{5}-\frac{666478}{1187}a^{4}+\frac{7683495}{87838}a^{3}+\frac{29753161}{87838}a^{2}-\frac{8105723}{87838}a-\frac{1393258}{43919}$, $\frac{31}{87838}a^{17}+\frac{31}{43919}a^{16}-\frac{967}{43919}a^{15}-\frac{1148}{43919}a^{14}+\frac{47871}{87838}a^{13}+\frac{15669}{43919}a^{12}-\frac{304652}{43919}a^{11}-\frac{186985}{87838}a^{10}+\frac{57900}{1187}a^{9}+\frac{209913}{43919}a^{8}-\frac{16516515}{87838}a^{7}-\frac{74620}{43919}a^{6}+\frac{32253367}{87838}a^{5}+\frac{1249543}{87838}a^{4}-\frac{13097267}{43919}a^{3}-\frac{5058859}{87838}a^{2}+\frac{5066091}{87838}a+\frac{1099793}{43919}$, $\frac{6}{43919}a^{17}-\frac{44}{43919}a^{16}-\frac{237}{43919}a^{15}+\frac{1804}{43919}a^{14}+\frac{7603}{87838}a^{13}-\frac{57323}{87838}a^{12}-\frac{33327}{43919}a^{11}+\frac{219346}{43919}a^{10}+\frac{186003}{43919}a^{9}-\frac{799868}{43919}a^{8}-\frac{1400757}{87838}a^{7}+\frac{2100145}{87838}a^{6}+\frac{3126889}{87838}a^{5}+\frac{420000}{43919}a^{4}-\frac{1487196}{43919}a^{3}-\frac{865362}{43919}a^{2}+\frac{1144019}{87838}a+\frac{97343}{43919}$, $\frac{1}{1187}a^{17}-\frac{49}{43919}a^{16}-\frac{3853}{87838}a^{15}+\frac{4985}{87838}a^{14}+\frac{41088}{43919}a^{13}-\frac{51610}{43919}a^{12}-\frac{460535}{43919}a^{11}+\frac{556003}{43919}a^{10}+\frac{5776143}{87838}a^{9}-\frac{6591733}{87838}a^{8}-\frac{19851393}{87838}a^{7}+\frac{10401332}{43919}a^{6}+\frac{16787371}{43919}a^{5}-\frac{15326162}{43919}a^{4}-\frac{20803293}{87838}a^{3}+\frac{7502752}{43919}a^{2}+\frac{502151}{43919}a-\frac{273327}{43919}$, $\frac{68}{43919}a^{17}+\frac{291}{87838}a^{16}-\frac{6583}{87838}a^{15}-\frac{13887}{87838}a^{14}+\frac{129635}{87838}a^{13}+\frac{133554}{43919}a^{12}-\frac{1336583}{87838}a^{11}-\frac{1324458}{43919}a^{10}+\frac{3878772}{43919}a^{9}+\frac{14388731}{87838}a^{8}-\frac{12766727}{43919}a^{7}-\frac{21017156}{43919}a^{6}+\frac{624734}{1187}a^{5}+\frac{60470767}{87838}a^{4}-\frac{42507725}{87838}a^{3}-\frac{34651169}{87838}a^{2}+\frac{14642819}{87838}a+\frac{1861657}{43919}$ (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: | \( 4187524895.96 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 4187524895.96 \cdot 1}{2\cdot\sqrt{6966114824903498893840251683313}}\cr\approx \mathstrut & 0.207956267514 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 18 |
The 18 conjugacy class representatives for $C_{18}$ |
Character table for $C_{18}$ |
Intermediate fields
\(\Q(\sqrt{33}) \), \(\Q(\zeta_{9})^+\), 6.6.26198073.1, \(\Q(\zeta_{27})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | R | $18$ | $18$ | R | $18$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | $18$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | $18$ |
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 $18$ | $18$ | $1$ | $45$ | |||
\(11\) | 11.18.9.1 | $x^{18} - 175384539 x^{4} + 1714871048 x^{2} - 21221529219$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |