Properties

Label 18.18.696...313.1
Degree $18$
Signature $[18, 0]$
Discriminant $6.966\times 10^{30}$
Root discriminant \(51.70\)
Ramified primes $3,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(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 + 531441*x^2 - 99144*x - 40553)
 
gp: K = bnfinit(y^18 - 54*y^16 + 1215*y^14 - 14742*y^12 + 104247*y^10 - 136*y^9 - 433026*y^8 + 3672*y^7 + 1010394*y^6 - 33048*y^5 - 1180980*y^4 + 110160*y^3 + 531441*y^2 - 99144*y - 40553, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(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 + 531441*x^2 - 99144*x - 40553);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(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 + 531441*x^2 - 99144*x - 40553)
 

\( x^{18} - 54 x^{16} + 1215 x^{14} - 14742 x^{12} + 104247 x^{10} - 136 x^{9} - 433026 x^{8} + 3672 x^{7} + 1010394 x^{6} - 33048 x^{5} - 1180980 x^{4} + 110160 x^{3} + \cdots - 40553 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

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)
 
Discriminant:   \(6966114824903498893840251683313\) \(\medspace = 3^{45}\cdot 11^{9}\) Copy content Toggle raw display
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:   \(3\), \(11\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{74}a^{9}-\frac{27}{74}a^{7}+\frac{21}{74}a^{5}+\frac{2}{37}a^{3}-\frac{11}{74}a-\frac{31}{74}$, $\frac{1}{74}a^{10}-\frac{27}{74}a^{8}+\frac{21}{74}a^{6}+\frac{2}{37}a^{4}-\frac{11}{74}a^{2}-\frac{31}{74}a$, $\frac{1}{74}a^{11}+\frac{16}{37}a^{7}-\frac{21}{74}a^{5}+\frac{23}{74}a^{3}-\frac{31}{74}a^{2}-\frac{1}{74}a-\frac{23}{74}$, $\frac{1}{74}a^{12}+\frac{16}{37}a^{8}-\frac{21}{74}a^{6}+\frac{23}{74}a^{4}-\frac{31}{74}a^{3}-\frac{1}{74}a^{2}-\frac{23}{74}a$, $\frac{1}{74}a^{13}+\frac{29}{74}a^{7}+\frac{17}{74}a^{5}-\frac{31}{74}a^{4}+\frac{19}{74}a^{3}-\frac{23}{74}a^{2}-\frac{9}{37}a+\frac{15}{37}$, $\frac{1}{87838}a^{14}-\frac{206}{43919}a^{13}-\frac{21}{43919}a^{12}-\frac{275}{43919}a^{11}-\frac{247}{43919}a^{10}-\frac{59}{87838}a^{9}+\frac{26379}{87838}a^{8}-\frac{32099}{87838}a^{7}-\frac{1113}{87838}a^{6}-\frac{15937}{43919}a^{5}+\frac{31901}{87838}a^{4}+\frac{26445}{87838}a^{3}-\frac{4099}{43919}a^{2}+\frac{9015}{87838}a-\frac{41171}{87838}$, $\frac{1}{87838}a^{15}-\frac{45}{87838}a^{13}-\frac{49}{87838}a^{12}-\frac{377}{87838}a^{11}+\frac{577}{87838}a^{10}-\frac{303}{87838}a^{9}+\frac{8235}{87838}a^{8}-\frac{9076}{43919}a^{7}-\frac{2573}{87838}a^{6}-\frac{5203}{87838}a^{5}-\frac{35}{2374}a^{4}-\frac{32071}{87838}a^{3}+\frac{18471}{43919}a^{2}+\frac{2805}{87838}a+\frac{29453}{87838}$, $\frac{1}{87838}a^{16}+\frac{403}{87838}a^{13}+\frac{107}{87838}a^{12}-\frac{433}{87838}a^{11}+\frac{10}{43919}a^{10}-\frac{355}{87838}a^{9}+\frac{10537}{43919}a^{8}+\frac{23665}{87838}a^{7}+\frac{17119}{87838}a^{6}+\frac{10412}{43919}a^{5}-\frac{3341}{43919}a^{4}+\frac{16227}{87838}a^{3}-\frac{16279}{43919}a^{2}-\frac{2031}{43919}a+\frac{11976}{43919}$, $\frac{1}{87838}a^{17}-\frac{1}{2374}a^{13}-\frac{125}{87838}a^{12}-\frac{299}{87838}a^{11}+\frac{249}{43919}a^{10}-\frac{255}{87838}a^{9}+\frac{10643}{87838}a^{8}-\frac{17569}{43919}a^{7}-\frac{3312}{43919}a^{6}+\frac{24875}{87838}a^{5}+\frac{4106}{43919}a^{4}+\frac{1895}{43919}a^{3}+\frac{18928}{43919}a^{2}+\frac{39745}{87838}a+\frac{7729}{43919}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{136}{43919}a^{15}-\frac{6120}{43919}a^{13}-\frac{729}{43919}a^{12}+\frac{110160}{43919}a^{11}+\frac{26244}{43919}a^{10}-\frac{1009800}{43919}a^{9}-\frac{354294}{43919}a^{8}+\frac{4957200}{43919}a^{7}+\frac{2205683}{43919}a^{6}-\frac{12492144}{43919}a^{5}-\frac{6221511}{43919}a^{4}+\frac{13870664}{43919}a^{3}+\frac{6473439}{43919}a^{2}-\frac{4376016}{43919}a-\frac{1126980}{43919}$, $\frac{80}{43919}a^{15}-\frac{3600}{43919}a^{13}-\frac{359}{43919}a^{12}+\frac{64800}{43919}a^{11}+\frac{12924}{43919}a^{10}-\frac{594000}{43919}a^{9}-\frac{174474}{43919}a^{8}+\frac{2916000}{43919}a^{7}+\frac{1085616}{43919}a^{6}-\frac{7348320}{43919}a^{5}-\frac{3053295}{43919}a^{4}+\frac{8164800}{43919}a^{3}+\frac{3140532}{43919}a^{2}-\frac{2624400}{43919}a-\frac{523422}{43919}$, $\frac{31}{87838}a^{17}+\frac{117}{87838}a^{16}-\frac{1741}{87838}a^{15}-\frac{5697}{87838}a^{14}+\frac{40537}{87838}a^{13}+\frac{56816}{43919}a^{12}-\frac{253313}{43919}a^{11}-\frac{597875}{43919}a^{10}+\frac{3673689}{87838}a^{9}+\frac{3558534}{43919}a^{8}-\frac{15558173}{87838}a^{7}-\frac{23773113}{87838}a^{6}+\frac{18265951}{43919}a^{5}+\frac{41055443}{87838}a^{4}-\frac{41104533}{87838}a^{3}-\frac{29227421}{87838}a^{2}+\frac{14678451}{87838}a+\frac{1918906}{43919}$, $\frac{31}{87838}a^{17}+\frac{253}{87838}a^{16}-\frac{1149}{87838}a^{15}-\frac{12063}{87838}a^{14}+\frac{13625}{87838}a^{13}+\frac{115615}{43919}a^{12}-\frac{505}{2374}a^{11}-\frac{1135922}{43919}a^{10}-\frac{404430}{43919}a^{9}+\frac{6031302}{43919}a^{8}+\frac{85149}{1187}a^{7}-\frac{901883}{2374}a^{6}-\frac{8660887}{43919}a^{5}+\frac{41993515}{87838}a^{4}+\frac{7769579}{43919}a^{3}-\frac{8461671}{43919}a^{2}-\frac{993267}{87838}a+\frac{215425}{43919}$, $\frac{105}{87838}a^{17}-\frac{68}{43919}a^{16}-\frac{4923}{87838}a^{15}+\frac{2924}{43919}a^{14}+\frac{47115}{43919}a^{13}-\frac{50456}{43919}a^{12}-\frac{474174}{43919}a^{11}+\frac{447440}{43919}a^{10}+\frac{5383179}{87838}a^{9}-\frac{4328509}{87838}a^{8}-\frac{463085}{2374}a^{7}+\frac{5534723}{43919}a^{6}+\frac{28355951}{87838}a^{5}-\frac{12834109}{87838}a^{4}-\frac{19959359}{87838}a^{3}+\frac{3791851}{87838}a^{2}+\frac{1647490}{43919}a+\frac{210346}{43919}$, $\frac{117}{87838}a^{16}-\frac{2808}{43919}a^{14}+\frac{54756}{43919}a^{12}-\frac{884}{43919}a^{11}-\frac{555984}{43919}a^{10}+\frac{29172}{43919}a^{9}+\frac{3127410}{43919}a^{8}-\frac{701315}{87838}a^{7}-\frac{9552816}{43919}a^{6}+\frac{3700599}{87838}a^{5}+\frac{14329224}{43919}a^{4}-\frac{4013001}{43919}a^{3}-\frac{16391687}{87838}a^{2}+\frac{4950207}{87838}a+\frac{813930}{43919}$, $\frac{599}{87838}a^{14}-\frac{1079}{87838}a^{13}-\frac{12579}{43919}a^{12}+\frac{42081}{87838}a^{11}+\frac{415107}{87838}a^{10}-\frac{631215}{87838}a^{9}-\frac{1698165}{43919}a^{8}+\frac{2272374}{43919}a^{7}+\frac{7132293}{43919}a^{6}-\frac{7953309}{43919}a^{5}-\frac{14264586}{43919}a^{4}+\frac{23859927}{87838}a^{3}+\frac{21396879}{87838}a^{2}-\frac{10225683}{87838}a-\frac{1310013}{43919}$, $\frac{340}{43919}a^{14}-\frac{1215}{87838}a^{13}-\frac{14280}{43919}a^{12}+\frac{47385}{87838}a^{11}+\frac{235620}{43919}a^{10}-\frac{710775}{87838}a^{9}-\frac{1927800}{43919}a^{8}+\frac{2558790}{43919}a^{7}+\frac{8096760}{43919}a^{6}-\frac{483935}{2374}a^{5}-\frac{32388227}{87838}a^{4}+\frac{13389135}{43919}a^{3}+\frac{12152262}{43919}a^{2}-\frac{5623740}{43919}a-\frac{1497843}{43919}$, $\frac{117}{87838}a^{16}-\frac{759}{87838}a^{15}-\frac{2468}{43919}a^{14}+\frac{16470}{43919}a^{13}+\frac{84907}{87838}a^{12}-\frac{569173}{87838}a^{11}-\frac{391554}{43919}a^{10}+\frac{2491572}{43919}a^{9}+\frac{2160675}{43919}a^{8}-\frac{23249285}{87838}a^{7}-\frac{7436016}{43919}a^{6}+\frac{27756095}{43919}a^{5}+\frac{14953748}{43919}a^{4}-\frac{29355636}{43919}a^{3}-\frac{26685503}{87838}a^{2}+\frac{9300861}{43919}a+\frac{2199282}{43919}$, $\frac{68}{43919}a^{17}-\frac{253}{87838}a^{16}-\frac{5745}{87838}a^{15}+\frac{5084}{43919}a^{14}+\frac{47726}{43919}a^{13}-\frac{159947}{87838}a^{12}-\frac{393304}{43919}a^{11}+\frac{1249299}{87838}a^{10}+\frac{3310005}{87838}a^{9}-\frac{5103109}{87838}a^{8}-\frac{3192309}{43919}a^{7}+\frac{10704739}{87838}a^{6}+\frac{1572661}{43919}a^{5}-\frac{5476801}{43919}a^{4}+\frac{3688537}{87838}a^{3}+\frac{4612533}{87838}a^{2}-\frac{2795113}{87838}a+\frac{43982}{43919}$, $\frac{55}{87838}a^{17}+\frac{155}{87838}a^{16}-\frac{2989}{87838}a^{15}-\frac{7721}{87838}a^{14}+\frac{33909}{43919}a^{13}+\frac{157589}{87838}a^{12}-\frac{832703}{87838}a^{11}-\frac{45811}{2374}a^{10}+\frac{2993964}{43919}a^{9}+\frac{10273045}{87838}a^{8}-\frac{25396941}{87838}a^{7}-\frac{34647037}{87838}a^{6}+\frac{60191875}{87838}a^{5}+\frac{59427871}{87838}a^{4}-\frac{34131640}{43919}a^{3}-\frac{40825129}{87838}a^{2}+\frac{23572155}{87838}a+\frac{2622215}{43919}$, $\frac{31}{87838}a^{17}-\frac{1989}{87838}a^{15}+\frac{340}{43919}a^{14}+\frac{25173}{43919}a^{13}-\frac{26373}{87838}a^{12}-\frac{653049}{87838}a^{11}+\frac{390707}{87838}a^{10}+\frac{2333205}{43919}a^{9}-\frac{1369344}{43919}a^{8}-\frac{9103023}{43919}a^{7}+\frac{4504578}{43919}a^{6}+\frac{35706523}{87838}a^{5}-\frac{5646172}{43919}a^{4}-\frac{28708229}{87838}a^{3}+\frac{618591}{43919}a^{2}+\frac{2866743}{43919}a+\frac{630270}{43919}$, $\frac{1}{1187}a^{17}-\frac{141}{87838}a^{16}-\frac{3535}{87838}a^{15}+\frac{6985}{87838}a^{14}+\frac{34060}{43919}a^{13}-\frac{71059}{43919}a^{12}-\frac{336505}{43919}a^{11}+\frac{762247}{43919}a^{10}+\frac{3550845}{87838}a^{9}-\frac{9177901}{87838}a^{8}-\frac{4589236}{43919}a^{7}+\frac{15167269}{43919}a^{6}+\frac{7327323}{87838}a^{5}-\frac{666478}{1187}a^{4}+\frac{7683495}{87838}a^{3}+\frac{29753161}{87838}a^{2}-\frac{8105723}{87838}a-\frac{1393258}{43919}$, $\frac{31}{87838}a^{17}+\frac{31}{43919}a^{16}-\frac{967}{43919}a^{15}-\frac{1148}{43919}a^{14}+\frac{47871}{87838}a^{13}+\frac{15669}{43919}a^{12}-\frac{304652}{43919}a^{11}-\frac{186985}{87838}a^{10}+\frac{57900}{1187}a^{9}+\frac{209913}{43919}a^{8}-\frac{16516515}{87838}a^{7}-\frac{74620}{43919}a^{6}+\frac{32253367}{87838}a^{5}+\frac{1249543}{87838}a^{4}-\frac{13097267}{43919}a^{3}-\frac{5058859}{87838}a^{2}+\frac{5066091}{87838}a+\frac{1099793}{43919}$, $\frac{6}{43919}a^{17}-\frac{44}{43919}a^{16}-\frac{237}{43919}a^{15}+\frac{1804}{43919}a^{14}+\frac{7603}{87838}a^{13}-\frac{57323}{87838}a^{12}-\frac{33327}{43919}a^{11}+\frac{219346}{43919}a^{10}+\frac{186003}{43919}a^{9}-\frac{799868}{43919}a^{8}-\frac{1400757}{87838}a^{7}+\frac{2100145}{87838}a^{6}+\frac{3126889}{87838}a^{5}+\frac{420000}{43919}a^{4}-\frac{1487196}{43919}a^{3}-\frac{865362}{43919}a^{2}+\frac{1144019}{87838}a+\frac{97343}{43919}$, $\frac{1}{1187}a^{17}-\frac{49}{43919}a^{16}-\frac{3853}{87838}a^{15}+\frac{4985}{87838}a^{14}+\frac{41088}{43919}a^{13}-\frac{51610}{43919}a^{12}-\frac{460535}{43919}a^{11}+\frac{556003}{43919}a^{10}+\frac{5776143}{87838}a^{9}-\frac{6591733}{87838}a^{8}-\frac{19851393}{87838}a^{7}+\frac{10401332}{43919}a^{6}+\frac{16787371}{43919}a^{5}-\frac{15326162}{43919}a^{4}-\frac{20803293}{87838}a^{3}+\frac{7502752}{43919}a^{2}+\frac{502151}{43919}a-\frac{273327}{43919}$, $\frac{68}{43919}a^{17}+\frac{291}{87838}a^{16}-\frac{6583}{87838}a^{15}-\frac{13887}{87838}a^{14}+\frac{129635}{87838}a^{13}+\frac{133554}{43919}a^{12}-\frac{1336583}{87838}a^{11}-\frac{1324458}{43919}a^{10}+\frac{3878772}{43919}a^{9}+\frac{14388731}{87838}a^{8}-\frac{12766727}{43919}a^{7}-\frac{21017156}{43919}a^{6}+\frac{624734}{1187}a^{5}+\frac{60470767}{87838}a^{4}-\frac{42507725}{87838}a^{3}-\frac{34651169}{87838}a^{2}+\frac{14642819}{87838}a+\frac{1861657}{43919}$ Copy content Toggle raw display (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)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(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 + 531441*x^2 - 99144*x - 40553)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(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 + 531441*x^2 - 99144*x - 40553, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(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 + 531441*x^2 - 99144*x - 40553);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(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 + 531441*x^2 - 99144*x - 40553);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_{18}$ (as 18T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.9.0.1}{9} }^{2}$ 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.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $18$$18$$1$$45$
\(11\) Copy content Toggle raw display 11.18.9.1$x^{18} - 175384539 x^{4} + 1714871048 x^{2} - 21221529219$$2$$9$$9$$C_{18}$$[\ ]_{2}^{9}$