LMFDB - Number field 18.0.450283905890997363000000000000.5

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 516*x^12 + 1071*x^11 + 1557*x^10 - 14664*x^9 + 43227*x^8 - 82809*x^7 + 123129*x^6 - 113832*x^5 + 17064*x^4 + 45324*x^3 + 13500*x^2 + 8316*x + 4356)

gp: K = bnfinit(y^18 - 9*y^17 + 45*y^16 - 126*y^15 + 189*y^14 - 27*y^13 - 516*y^12 + 1071*y^11 + 1557*y^10 - 14664*y^9 + 43227*y^8 - 82809*y^7 + 123129*y^6 - 113832*y^5 + 17064*y^4 + 45324*y^3 + 13500*y^2 + 8316*y + 4356, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 516*x^12 + 1071*x^11 + 1557*x^10 - 14664*x^9 + 43227*x^8 - 82809*x^7 + 123129*x^6 - 113832*x^5 + 17064*x^4 + 45324*x^3 + 13500*x^2 + 8316*x + 4356);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 516*x^12 + 1071*x^11 + 1557*x^10 - 14664*x^9 + 43227*x^8 - 82809*x^7 + 123129*x^6 - 113832*x^5 + 17064*x^4 + 45324*x^3 + 13500*x^2 + 8316*x + 4356)

$ x^{18} - 9 x^{17} + 45 x^{16} - 126 x^{15} + 189 x^{14} - 27 x^{13} - 516 x^{12} + 1071 x^{11} + \cdots + 4356 $ x^18 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 516*x^12 + 1071*x^11 + 1557*x^10 - 14664*x^9 + 43227*x^8 - 82809*x^7 + 123129*x^6 - 113832*x^5 + 17064*x^4 + 45324*x^3 + 13500*x^2 + 8316*x + 4356

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:	$[0, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-450283905890997363000000000000$ -450283905890997363000000000000 $\medspace = -\,2^{12}\cdot 3^{37}\cdot 5^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$44.40$	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:	$2^{2/3}3^{37/18}5^{2/3}\approx 44.40336798307826$
Ramified primes:	$2$, $3$, $5$ 2, 3, 5	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-3}) $
$\card{ \Gal(K/\Q) }$:	$18$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{7}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{45}a^{9}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{15}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{1}{15}$, $\frac{1}{45}a^{10}+\frac{1}{5}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a-\frac{1}{5}$, $\frac{1}{45}a^{11}+\frac{4}{15}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{7}{15}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{450}a^{12}+\frac{2}{225}a^{11}-\frac{2}{225}a^{10}-\frac{1}{90}a^{9}-\frac{2}{25}a^{7}+\frac{7}{150}a^{6}+\frac{14}{75}a^{5}-\frac{7}{15}a^{4}+\frac{1}{30}a^{3}-\frac{4}{75}a^{2}-\frac{11}{75}a-\frac{4}{75}$, $\frac{1}{2700}a^{13}-\frac{1}{900}a^{12}+\frac{1}{150}a^{11}-\frac{1}{100}a^{10}+\frac{1}{180}a^{9}-\frac{2}{25}a^{8}-\frac{59}{900}a^{7}+\frac{1}{100}a^{6}-\frac{31}{150}a^{5}+\frac{53}{180}a^{4}+\frac{49}{300}a^{3}-\frac{6}{25}a^{2}+\frac{17}{50}a+\frac{31}{150}$, $\frac{1}{2700}a^{14}-\frac{1}{900}a^{12}-\frac{7}{900}a^{11}-\frac{1}{150}a^{10}+\frac{1}{300}a^{9}+\frac{17}{180}a^{8}-\frac{2}{75}a^{7}-\frac{7}{100}a^{6}-\frac{89}{900}a^{5}-\frac{21}{50}a^{4}+\frac{7}{60}a^{3}+\frac{19}{150}a^{2}-\frac{2}{25}a+\frac{13}{50}$, $\frac{1}{491400}a^{15}+\frac{43}{491400}a^{14}+\frac{1}{10920}a^{13}+\frac{1}{20475}a^{12}+\frac{193}{32760}a^{11}+\frac{1669}{163800}a^{10}-\frac{59}{6825}a^{9}-\frac{1513}{32760}a^{8}-\frac{4621}{54600}a^{7}-\frac{493}{8190}a^{6}+\frac{81691}{163800}a^{5}+\frac{1129}{4200}a^{4}-\frac{125}{312}a^{3}-\frac{5387}{27300}a^{2}+\frac{577}{5460}a-\frac{5489}{27300}$, $\frac{1}{982800}a^{16}-\frac{1}{982800}a^{15}-\frac{1}{36400}a^{14}-\frac{17}{122850}a^{13}-\frac{23}{65520}a^{12}+\frac{599}{109200}a^{11}+\frac{67}{8190}a^{10}-\frac{389}{36400}a^{9}-\frac{281}{5200}a^{8}-\frac{269}{5460}a^{7}+\frac{19763}{327600}a^{6}+\frac{10513}{21840}a^{5}+\frac{19031}{327600}a^{4}+\frac{281}{18200}a^{3}-\frac{1769}{3640}a^{2}+\frac{4633}{18200}a+\frac{5597}{13650}$, $\frac{1}{53\!\cdots\!00}a^{17}+\frac{1810860823}{53\!\cdots\!00}a^{16}+\frac{1893923791}{53\!\cdots\!00}a^{15}+\frac{21553673837}{26\!\cdots\!00}a^{14}-\frac{24307891321}{152357965634160}a^{13}+\frac{343092561479}{592503199688400}a^{12}-\frac{4719475837261}{888754799532600}a^{11}-\frac{2705797562153}{253929942723600}a^{10}+\frac{4500814941499}{592503199688400}a^{9}+\frac{1134149756321}{25392994272360}a^{8}-\frac{7888085742499}{136731507620400}a^{7}+\frac{113580175230727}{17\!\cdots\!00}a^{6}-\frac{169809727697399}{17\!\cdots\!00}a^{5}+\frac{85301523097897}{444377399766300}a^{4}+\frac{14558309510611}{29625159984420}a^{3}-\frac{2330361425611}{14107219040200}a^{2}-\frac{15192830396421}{49375266640700}a+\frac{903761133139}{1923711687300}$ 1, a, a^2, a^3, a^4, a^5, 1/5*a^6 + 2/5*a^5 + 1/5*a^4 - 2/5*a^3 + 1/5*a^2 + 2/5*a + 1/5, 1/5*a^7 + 2/5*a^5 + 1/5*a^4 + 2/5*a - 2/5, 1/5*a^8 + 2/5*a^5 - 2/5*a^4 - 1/5*a^3 - 1/5*a - 2/5, 1/45*a^9 + 1/5*a^5 - 2/5*a^4 - 2/15*a^3 - 2/5*a^2 + 1/5*a + 1/15, 1/45*a^10 + 1/5*a^5 - 1/3*a^4 - 1/3*a - 1/5, 1/45*a^11 + 4/15*a^5 - 1/5*a^4 + 2/5*a^3 + 7/15*a^2 + 2/5*a - 1/5, 1/450*a^12 + 2/225*a^11 - 2/225*a^10 - 1/90*a^9 - 2/25*a^7 + 7/150*a^6 + 14/75*a^5 - 7/15*a^4 + 1/30*a^3 - 4/75*a^2 - 11/75*a - 4/75, 1/2700*a^13 - 1/900*a^12 + 1/150*a^11 - 1/100*a^10 + 1/180*a^9 - 2/25*a^8 - 59/900*a^7 + 1/100*a^6 - 31/150*a^5 + 53/180*a^4 + 49/300*a^3 - 6/25*a^2 + 17/50*a + 31/150, 1/2700*a^14 - 1/900*a^12 - 7/900*a^11 - 1/150*a^10 + 1/300*a^9 + 17/180*a^8 - 2/75*a^7 - 7/100*a^6 - 89/900*a^5 - 21/50*a^4 + 7/60*a^3 + 19/150*a^2 - 2/25*a + 13/50, 1/491400*a^15 + 43/491400*a^14 + 1/10920*a^13 + 1/20475*a^12 + 193/32760*a^11 + 1669/163800*a^10 - 59/6825*a^9 - 1513/32760*a^8 - 4621/54600*a^7 - 493/8190*a^6 + 81691/163800*a^5 + 1129/4200*a^4 - 125/312*a^3 - 5387/27300*a^2 + 577/5460*a - 5489/27300, 1/982800*a^16 - 1/982800*a^15 - 1/36400*a^14 - 17/122850*a^13 - 23/65520*a^12 + 599/109200*a^11 + 67/8190*a^10 - 389/36400*a^9 - 281/5200*a^8 - 269/5460*a^7 + 19763/327600*a^6 + 10513/21840*a^5 + 19031/327600*a^4 + 281/18200*a^3 - 1769/3640*a^2 + 4633/18200*a + 5597/13650, 1/5332528797195600*a^17 + 1810860823/5332528797195600*a^16 + 1893923791/5332528797195600*a^15 + 21553673837/2666264398597800*a^14 - 24307891321/152357965634160*a^13 + 343092561479/592503199688400*a^12 - 4719475837261/888754799532600*a^11 - 2705797562153/253929942723600*a^10 + 4500814941499/592503199688400*a^9 + 1134149756321/25392994272360*a^8 - 7888085742499/136731507620400*a^7 + 113580175230727/1777509599065200*a^6 - 169809727697399/1777509599065200*a^5 + 85301523097897/444377399766300*a^4 + 14558309510611/29625159984420*a^3 - 2330361425611/14107219040200*a^2 - 15192830396421/49375266640700*a + 903761133139/1923711687300

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (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:	$8$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{2330596735}{30471593126832} a^{17} + \frac{8337083525}{11850063993768} a^{16} - \frac{42205415257}{11850063993768} a^{15} + \frac{242116368005}{23700127987536} a^{14} - \frac{1137037775285}{71100383962608} a^{13} + \frac{2885146570}{634824856809} a^{12} + \frac{2705471032685}{71100383962608} a^{11} - \frac{6092379038987}{71100383962608} a^{10} - \frac{42410881345}{390661450344} a^{9} + \frac{81252868483685}{71100383962608} a^{8} - \frac{11756117569715}{3385732569648} a^{7} + \frac{40567998752245}{5925031996884} a^{6} - \frac{5198272583684}{493752666407} a^{5} + \frac{251078784934655}{23700127987536} a^{4} - \frac{10210731386615}{2962515998442} a^{3} - \frac{3430466263275}{1975010665628} a^{2} - \frac{2955645331495}{1692866284824} a - \frac{816500707}{134659818111} $ -(2330596735)/(30471593126832)a^(17) + (8337083525)/(11850063993768)a^(16) - (42205415257)/(11850063993768)a^(15) + (242116368005)/(23700127987536)a^(14) - (1137037775285)/(71100383962608)a^(13) + (2885146570)/(634824856809)a^(12) + (2705471032685)/(71100383962608)a^(11) - (6092379038987)/(71100383962608)a^(10) - (42410881345)/(390661450344)a^(9) + (81252868483685)/(71100383962608)a^(8) - (11756117569715)/(3385732569648)a^(7) + (40567998752245)/(5925031996884)a^(6) - (5198272583684)/(493752666407)a^(5) + (251078784934655)/(23700127987536)a^(4) - (10210731386615)/(2962515998442)a^(3) - (3430466263275)/(1975010665628)a^(2) - (2955645331495)/(1692866284824)*a - (816500707)/(134659818111) (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)
Fundamental units:	$\frac{37997493239}{533252879719560}a^{17}-\frac{702544588199}{10\!\cdots\!20}a^{16}+\frac{17762491985027}{53\!\cdots\!00}a^{15}-\frac{10175326550981}{10\!\cdots\!20}a^{14}+\frac{2608773727279}{177750959906520}a^{13}-\frac{124021750791}{39500213312560}a^{12}-\frac{22775610797587}{592503199688400}a^{11}+\frac{8199726914629}{98750533281400}a^{10}+\frac{2452541711149}{23700127987536}a^{9}-\frac{385121986158673}{355501919813040}a^{8}+\frac{82913522530757}{25392994272360}a^{7}-\frac{11\!\cdots\!09}{17\!\cdots\!00}a^{6}+\frac{16\!\cdots\!03}{17\!\cdots\!00}a^{5}-\frac{10\!\cdots\!41}{118500639937680}a^{4}+\frac{13203502117177}{8464331424120}a^{3}+\frac{47308594150133}{11850063993768}a^{2}+\frac{32475279263489}{296251599844200}a-\frac{74881257911}{73988911050}$, $\frac{57694763317}{10\!\cdots\!20}a^{17}-\frac{1259207950277}{26\!\cdots\!00}a^{16}+\frac{2982028465421}{13\!\cdots\!00}a^{15}-\frac{5993185172687}{10\!\cdots\!20}a^{14}+\frac{2084162132447}{355501919813040}a^{13}+\frac{32293550243}{4232165712060}a^{12}-\frac{1640963234641}{45577169206800}a^{11}+\frac{9416976463053}{197501066562800}a^{10}+\frac{7722681406511}{59250319968840}a^{9}-\frac{289772027508727}{355501919813040}a^{8}+\frac{706229751303061}{355501919813040}a^{7}-\frac{333209224702381}{111094349941575}a^{6}+\frac{29\!\cdots\!49}{888754799532600}a^{5}-\frac{11321323347757}{23700127987536}a^{4}-\frac{361571434565333}{59250319968840}a^{3}+\frac{766139284807}{113942923017}a^{2}+\frac{446218519602203}{296251599844200}a-\frac{1108706267777}{2693196362220}$, $\frac{4539051991}{133313219929890}a^{17}-\frac{375192172577}{13\!\cdots\!00}a^{16}+\frac{116805369887}{88875479953260}a^{15}-\frac{1095184017869}{333283049824725}a^{14}+\frac{1691124067163}{444377399766300}a^{13}+\frac{921954341603}{444377399766300}a^{12}-\frac{398805407857}{24687633320350}a^{11}+\frac{633712819219}{29625159984420}a^{10}+\frac{33288777360287}{444377399766300}a^{9}-\frac{47782251416932}{111094349941575}a^{8}+\frac{69900599222671}{63482485680900}a^{7}-\frac{294735106987613}{148125799922100}a^{6}+\frac{8502275731793}{3174124284045}a^{5}-\frac{71241922431356}{37031449980525}a^{4}-\frac{14584375724546}{37031449980525}a^{3}+\frac{54397927494059}{37031449980525}a^{2}-\frac{1044776143673}{7406289996105}a+\frac{67539621278}{673299090555}$, $\frac{27880523329}{10\!\cdots\!20}a^{17}-\frac{311770704811}{26\!\cdots\!00}a^{16}+\frac{1768641071}{5078598854472}a^{15}+\frac{1180376735959}{53\!\cdots\!00}a^{14}-\frac{245498770901}{118500639937680}a^{13}+\frac{604116400339}{148125799922100}a^{12}-\frac{453311684183}{253929942723600}a^{11}-\frac{23029440264053}{17\!\cdots\!00}a^{10}+\frac{7200546493489}{98750533281400}a^{9}-\frac{1770744251753}{16928662848240}a^{8}-\frac{88682537442593}{17\!\cdots\!00}a^{7}+\frac{7773367121162}{37031449980525}a^{6}-\frac{155236690734679}{222188699883150}a^{5}+\frac{302279889117287}{197501066562800}a^{4}-\frac{279281672435}{987505332814}a^{3}-\frac{289736851194211}{148125799922100}a^{2}-\frac{7315259400941}{19750106656280}a-\frac{79863972082}{1122165150925}$, $\frac{820235610851}{26\!\cdots\!00}a^{17}-\frac{969248200633}{333283049824725}a^{16}+\frac{900849548321}{59250319968840}a^{15}-\frac{1178358021277}{25637157678825}a^{14}+\frac{53230280062117}{666566099649450}a^{13}-\frac{19228077952537}{444377399766300}a^{12}-\frac{69122697615989}{444377399766300}a^{11}+\frac{49538555591134}{111094349941575}a^{10}+\frac{3929999127631}{15870621420225}a^{9}-\frac{58951860221318}{12343816660175}a^{8}+\frac{17\!\cdots\!53}{111094349941575}a^{7}-\frac{125813119433283}{3798097433900}a^{6}+\frac{45\!\cdots\!63}{888754799532600}a^{5}-\frac{58\!\cdots\!47}{111094349941575}a^{4}+\frac{19\!\cdots\!19}{98750533281400}a^{3}+\frac{554194072000801}{49375266640700}a^{2}-\frac{47422653140419}{37031449980525}a+\frac{7083029690171}{1923711687300}$, $\frac{811524515891}{17\!\cdots\!00}a^{17}-\frac{7757183075959}{17\!\cdots\!00}a^{16}+\frac{40256472387979}{17\!\cdots\!00}a^{15}-\frac{9979970277337}{148125799922100}a^{14}+\frac{591688815156863}{53\!\cdots\!00}a^{13}-\frac{76248712387831}{17\!\cdots\!00}a^{12}-\frac{109127217536873}{444377399766300}a^{11}+\frac{10\!\cdots\!47}{17\!\cdots\!00}a^{10}+\frac{886291304741189}{17\!\cdots\!00}a^{9}-\frac{40723412404403}{5697146150850}a^{8}+\frac{41\!\cdots\!11}{17\!\cdots\!00}a^{7}-\frac{27\!\cdots\!91}{592503199688400}a^{6}+\frac{20\!\cdots\!01}{28214438080400}a^{5}-\frac{65\!\cdots\!57}{888754799532600}a^{4}+\frac{68\!\cdots\!21}{296251599844200}a^{3}+\frac{18\!\cdots\!43}{59250319968840}a^{2}-\frac{50855219295721}{5290207140075}a-\frac{3715152151199}{480927921825}$, $\frac{152860106653}{53\!\cdots\!00}a^{17}-\frac{692128655063}{26\!\cdots\!00}a^{16}+\frac{277960398601}{222188699883150}a^{15}-\frac{17279394758719}{53\!\cdots\!00}a^{14}+\frac{22531931115377}{53\!\cdots\!00}a^{13}-\frac{2049121552}{4443773997663}a^{12}-\frac{1343018867143}{355501919813040}a^{11}+\frac{12919166240821}{17\!\cdots\!00}a^{10}+\frac{32592797227079}{888754799532600}a^{9}-\frac{15307679664241}{45577169206800}a^{8}+\frac{395365760858741}{355501919813040}a^{7}-\frac{62078806644143}{29625159984420}a^{6}+\frac{31\!\cdots\!77}{888754799532600}a^{5}-\frac{13\!\cdots\!89}{253929942723600}a^{4}+\frac{407416031084653}{98750533281400}a^{3}-\frac{14143955423021}{12343816660175}a^{2}+\frac{517307289379663}{296251599844200}a+\frac{16253632376927}{13465981811100}$, $\frac{8643833521}{53\!\cdots\!00}a^{17}+\frac{73555947541}{17\!\cdots\!00}a^{16}-\frac{225344927039}{761789828170800}a^{15}+\frac{3675938153009}{26\!\cdots\!00}a^{14}-\frac{14074277947007}{53\!\cdots\!00}a^{13}+\frac{3987393444889}{17\!\cdots\!00}a^{12}+\frac{4527775668083}{888754799532600}a^{11}-\frac{23839212099067}{17\!\cdots\!00}a^{10}+\frac{35020690738349}{17\!\cdots\!00}a^{9}+\frac{104437332793687}{888754799532600}a^{8}-\frac{106810083210797}{253929942723600}a^{7}+\frac{245613155235397}{253929942723600}a^{6}-\frac{24\!\cdots\!83}{17\!\cdots\!00}a^{5}+\frac{464178785312597}{222188699883150}a^{4}-\frac{1787220955783}{37031449980525}a^{3}-\frac{23301296769173}{98750533281400}a^{2}+\frac{316530782231}{148125799922100}a-\frac{1504252263949}{13465981811100}$ 37997493239/533252879719560a^17 - 702544588199/1066505759439120a^16 + 17762491985027/5332528797195600a^15 - 10175326550981/1066505759439120a^14 + 2608773727279/177750959906520a^13 - 124021750791/39500213312560a^12 - 22775610797587/592503199688400a^11 + 8199726914629/98750533281400a^10 + 2452541711149/23700127987536a^9 - 385121986158673/355501919813040a^8 + 82913522530757/25392994272360a^7 - 11276118485298109/1777509599065200a^6 + 16851973242871103/1777509599065200a^5 - 1062963936105041/118500639937680a^4 + 13203502117177/8464331424120a^3 + 47308594150133/11850063993768a^2 + 32475279263489/296251599844200a - 74881257911/73988911050, 57694763317/1066505759439120a^17 - 1259207950277/2666264398597800a^16 + 2982028465421/1333132199298900a^15 - 5993185172687/1066505759439120a^14 + 2084162132447/355501919813040a^13 + 32293550243/4232165712060a^12 - 1640963234641/45577169206800a^11 + 9416976463053/197501066562800a^10 + 7722681406511/59250319968840a^9 - 289772027508727/355501919813040a^8 + 706229751303061/355501919813040a^7 - 333209224702381/111094349941575a^6 + 2949493121283949/888754799532600a^5 - 11321323347757/23700127987536a^4 - 361571434565333/59250319968840a^3 + 766139284807/113942923017a^2 + 446218519602203/296251599844200a - 1108706267777/2693196362220, 4539051991/133313219929890a^17 - 375192172577/1333132199298900a^16 + 116805369887/88875479953260a^15 - 1095184017869/333283049824725a^14 + 1691124067163/444377399766300a^13 + 921954341603/444377399766300a^12 - 398805407857/24687633320350a^11 + 633712819219/29625159984420a^10 + 33288777360287/444377399766300a^9 - 47782251416932/111094349941575a^8 + 69900599222671/63482485680900a^7 - 294735106987613/148125799922100a^6 + 8502275731793/3174124284045a^5 - 71241922431356/37031449980525a^4 - 14584375724546/37031449980525a^3 + 54397927494059/37031449980525a^2 - 1044776143673/7406289996105a + 67539621278/673299090555, 27880523329/1066505759439120a^17 - 311770704811/2666264398597800a^16 + 1768641071/5078598854472a^15 + 1180376735959/5332528797195600a^14 - 245498770901/118500639937680a^13 + 604116400339/148125799922100a^12 - 453311684183/253929942723600a^11 - 23029440264053/1777509599065200a^10 + 7200546493489/98750533281400a^9 - 1770744251753/16928662848240a^8 - 88682537442593/1777509599065200a^7 + 7773367121162/37031449980525a^6 - 155236690734679/222188699883150a^5 + 302279889117287/197501066562800a^4 - 279281672435/987505332814a^3 - 289736851194211/148125799922100a^2 - 7315259400941/19750106656280a - 79863972082/1122165150925, 820235610851/2666264398597800a^17 - 969248200633/333283049824725a^16 + 900849548321/59250319968840a^15 - 1178358021277/25637157678825a^14 + 53230280062117/666566099649450a^13 - 19228077952537/444377399766300a^12 - 69122697615989/444377399766300a^11 + 49538555591134/111094349941575a^10 + 3929999127631/15870621420225a^9 - 58951860221318/12343816660175a^8 + 1770807374535253/111094349941575a^7 - 125813119433283/3798097433900a^6 + 45750778401552163/888754799532600a^5 - 5825189679734947/111094349941575a^4 + 1932234704682619/98750533281400a^3 + 554194072000801/49375266640700a^2 - 47422653140419/37031449980525a + 7083029690171/1923711687300, 811524515891/1777509599065200a^17 - 7757183075959/1777509599065200a^16 + 40256472387979/1777509599065200a^15 - 9979970277337/148125799922100a^14 + 591688815156863/5332528797195600a^13 - 76248712387831/1777509599065200a^12 - 109127217536873/444377399766300a^11 + 1072229166965147/1777509599065200a^10 + 886291304741189/1777509599065200a^9 - 40723412404403/5697146150850a^8 + 41081455127746811/1777509599065200a^7 - 27601392089394391/592503199688400a^6 + 2009057571879301/28214438080400a^5 - 65078309957342057/888754799532600a^4 + 6824507629849421/296251599844200a^3 + 1816714563313843/59250319968840a^2 - 50855219295721/5290207140075a - 3715152151199/480927921825, 152860106653/5332528797195600a^17 - 692128655063/2666264398597800a^16 + 277960398601/222188699883150a^15 - 17279394758719/5332528797195600a^14 + 22531931115377/5332528797195600a^13 - 2049121552/4443773997663a^12 - 1343018867143/355501919813040a^11 + 12919166240821/1777509599065200a^10 + 32592797227079/888754799532600a^9 - 15307679664241/45577169206800a^8 + 395365760858741/355501919813040a^7 - 62078806644143/29625159984420a^6 + 3108017913655777/888754799532600a^5 - 1307464885004989/253929942723600a^4 + 407416031084653/98750533281400a^3 - 14143955423021/12343816660175a^2 + 517307289379663/296251599844200a + 16253632376927/13465981811100, 8643833521/5332528797195600a^17 + 73555947541/1777509599065200a^16 - 225344927039/761789828170800a^15 + 3675938153009/2666264398597800a^14 - 14074277947007/5332528797195600a^13 + 3987393444889/1777509599065200a^12 + 4527775668083/888754799532600a^11 - 23839212099067/1777509599065200a^10 + 35020690738349/1777509599065200a^9 + 104437332793687/888754799532600a^8 - 106810083210797/253929942723600a^7 + 245613155235397/253929942723600a^6 - 2489313107661383/1777509599065200a^5 + 464178785312597/222188699883150a^4 - 1787220955783/37031449980525a^3 - 23301296769173/98750533281400a^2 + 316530782231/148125799922100a - 1504252263949/13465981811100 (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:	$ 8347574.07937 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 8347574.07937 \cdot 27}{6\cdot\sqrt{450283905890997363000000000000}}\cr\approx \mathstrut & 0.854374737569 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 516*x^12 + 1071*x^11 + 1557*x^10 - 14664*x^9 + 43227*x^8 - 82809*x^7 + 123129*x^6 - 113832*x^5 + 17064*x^4 + 45324*x^3 + 13500*x^2 + 8316*x + 4356)

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 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 516*x^12 + 1071*x^11 + 1557*x^10 - 14664*x^9 + 43227*x^8 - 82809*x^7 + 123129*x^6 - 113832*x^5 + 17064*x^4 + 45324*x^3 + 13500*x^2 + 8316*x + 4356, 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 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 516*x^12 + 1071*x^11 + 1557*x^10 - 14664*x^9 + 43227*x^8 - 82809*x^7 + 123129*x^6 - 113832*x^5 + 17064*x^4 + 45324*x^3 + 13500*x^2 + 8316*x + 4356);

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 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 516*x^12 + 1071*x^11 + 1557*x^10 - 14664*x^9 + 43227*x^8 - 82809*x^7 + 123129*x^6 - 113832*x^5 + 17064*x^4 + 45324*x^3 + 13500*x^2 + 8316*x + 4356);

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

$C_3:S_3$ (as 18T4):

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 solvable group of order 18

The 6 conjugacy class representatives for $C_3^2 : C_2$

Character table for $C_3^2 : C_2$

Intermediate fields

$\Q(\sqrt{-3}) $, 3.1.6075.1 x3, 3.1.108.1 x3, 3.1.24300.4 x3, 3.1.24300.3 x3, 6.0.110716875.1, 6.0.34992.1, 6.0.1771470000.2, 6.0.1771470000.1, 9.1.387420489000000.13 x9

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]

Sibling fields

Degree 9 sibling:	9.1.387420489000000.13
Minimal sibling:	9.1.387420489000000.13

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	${\href{/padicField/7.3.0.1}{3} }^{6}$	${\href{/padicField/11.2.0.1}{2} }^{9}$	${\href{/padicField/13.3.0.1}{3} }^{6}$	${\href{/padicField/17.2.0.1}{2} }^{9}$	${\href{/padicField/19.3.0.1}{3} }^{6}$	${\href{/padicField/23.2.0.1}{2} }^{9}$	${\href{/padicField/29.2.0.1}{2} }^{9}$	${\href{/padicField/31.3.0.1}{3} }^{6}$	${\href{/padicField/37.3.0.1}{3} }^{6}$	${\href{/padicField/41.2.0.1}{2} }^{9}$	${\href{/padicField/43.1.0.1}{1} }^{18}$	${\href{/padicField/47.2.0.1}{2} }^{9}$	${\href{/padicField/53.2.0.1}{2} }^{9}$	${\href{/padicField/59.2.0.1}{2} }^{9}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# 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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	2.6.4.1	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$3$ 3		Deg $18$	$18$	$1$	$37$
$5$ 5	5.6.4.1	$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	5.6.4.1	$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
	5.6.4.1	$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$

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