LMFDB - Number field 18.0.18283008447003871936512.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 3*x^16 + 8*x^15 - 4*x^13 + 40*x^12 - 74*x^11 - 3*x^10 + 299*x^9 - 181*x^8 - 158*x^7 + 506*x^6 + 166*x^5 + 86*x^4 + 36*x^3 + 9*x^2 + x + 1)

gp: K = bnfinit(y^18 - y^17 - 3*y^16 + 8*y^15 - 4*y^13 + 40*y^12 - 74*y^11 - 3*y^10 + 299*y^9 - 181*y^8 - 158*y^7 + 506*y^6 + 166*y^5 + 86*y^4 + 36*y^3 + 9*y^2 + y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 - 3*x^16 + 8*x^15 - 4*x^13 + 40*x^12 - 74*x^11 - 3*x^10 + 299*x^9 - 181*x^8 - 158*x^7 + 506*x^6 + 166*x^5 + 86*x^4 + 36*x^3 + 9*x^2 + x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 - 3*x^16 + 8*x^15 - 4*x^13 + 40*x^12 - 74*x^11 - 3*x^10 + 299*x^9 - 181*x^8 - 158*x^7 + 506*x^6 + 166*x^5 + 86*x^4 + 36*x^3 + 9*x^2 + x + 1)

$ x^{18} - x^{17} - 3 x^{16} + 8 x^{15} - 4 x^{13} + 40 x^{12} - 74 x^{11} - 3 x^{10} + 299 x^{9} + \cdots + 1 $ x^18 - x^17 - 3*x^16 + 8*x^15 - 4*x^13 + 40*x^12 - 74*x^11 - 3*x^10 + 299*x^9 - 181*x^8 - 158*x^7 + 506*x^6 + 166*x^5 + 86*x^4 + 36*x^3 + 9*x^2 + x + 1

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:	$-18283008447003871936512$ -18283008447003871936512 $\medspace = -\,2^{26}\cdot 3^{9}\cdot 7^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$17.25$	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:	$2^{11/6}3^{1/2}7^{2/3}\approx 22.58643281430495$
Ramified primes:	$2$, $3$, $7$ 2, 3, 7	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}) $
$\card{ \Aut(K/\Q) }$:	$6$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{6}a^{12}-\frac{1}{6}a^{10}-\frac{1}{3}a^{7}-\frac{1}{2}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a-\frac{1}{6}$, $\frac{1}{6}a^{13}-\frac{1}{6}a^{11}-\frac{1}{6}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{18}a^{14}-\frac{1}{18}a^{10}-\frac{1}{9}a^{9}-\frac{1}{18}a^{8}-\frac{2}{9}a^{7}+\frac{1}{18}a^{6}-\frac{1}{9}a^{5}-\frac{1}{2}a^{4}+\frac{4}{9}a^{2}+\frac{1}{9}a+\frac{7}{18}$, $\frac{1}{18}a^{15}-\frac{1}{18}a^{11}-\frac{1}{9}a^{10}-\frac{1}{18}a^{9}+\frac{1}{9}a^{8}+\frac{7}{18}a^{7}-\frac{1}{9}a^{6}-\frac{1}{2}a^{5}+\frac{1}{3}a^{4}+\frac{4}{9}a^{3}+\frac{1}{9}a^{2}+\frac{1}{18}a+\frac{1}{3}$, $\frac{1}{918}a^{16}+\frac{7}{918}a^{15}-\frac{11}{918}a^{14}-\frac{23}{306}a^{13}-\frac{1}{918}a^{12}+\frac{20}{153}a^{11}+\frac{73}{459}a^{10}-\frac{7}{918}a^{9}-\frac{35}{459}a^{8}+\frac{1}{27}a^{7}-\frac{86}{459}a^{6}+\frac{421}{918}a^{5}-\frac{265}{918}a^{4}-\frac{287}{918}a^{3}-\frac{397}{918}a^{2}-\frac{7}{17}a+\frac{361}{918}$, $\frac{1}{892443743838}a^{17}+\frac{116991155}{297481247946}a^{16}+\frac{917158316}{148740623973}a^{15}+\frac{94295551}{3734074242}a^{14}-\frac{206168921}{3734074242}a^{13}-\frac{37246972451}{892443743838}a^{12}+\frac{34480183991}{892443743838}a^{11}+\frac{764644733}{6466983651}a^{10}+\frac{36867991495}{297481247946}a^{9}-\frac{28510051646}{446221871919}a^{8}+\frac{355141804543}{892443743838}a^{7}-\frac{19754763872}{446221871919}a^{6}-\frac{27112564195}{446221871919}a^{5}+\frac{157817220161}{892443743838}a^{4}+\frac{10089029663}{38801901906}a^{3}+\frac{162334114921}{892443743838}a^{2}-\frac{162094802527}{446221871919}a-\frac{25895071651}{892443743838}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/3*a^8 + 1/3*a^7 + 1/3*a^4 - 1/3*a + 1/3, 1/3*a^9 - 1/3*a^7 + 1/3*a^5 - 1/3*a^4 - 1/3*a^2 - 1/3*a - 1/3, 1/3*a^10 + 1/3*a^7 + 1/3*a^6 - 1/3*a^5 + 1/3*a^4 - 1/3*a^3 - 1/3*a^2 + 1/3*a + 1/3, 1/3*a^11 - 1/3*a^6 + 1/3*a^5 + 1/3*a^4 - 1/3*a^3 + 1/3*a^2 - 1/3*a - 1/3, 1/6*a^12 - 1/6*a^10 - 1/3*a^7 - 1/2*a^6 + 1/3*a^5 - 1/3*a^4 + 1/3*a^3 - 1/2*a^2 - 1/3*a - 1/6, 1/6*a^13 - 1/6*a^11 - 1/6*a^7 + 1/3*a^6 - 1/3*a^5 - 1/3*a^4 - 1/2*a^3 - 1/3*a^2 - 1/2*a + 1/3, 1/18*a^14 - 1/18*a^10 - 1/9*a^9 - 1/18*a^8 - 2/9*a^7 + 1/18*a^6 - 1/9*a^5 - 1/2*a^4 + 4/9*a^2 + 1/9*a + 7/18, 1/18*a^15 - 1/18*a^11 - 1/9*a^10 - 1/18*a^9 + 1/9*a^8 + 7/18*a^7 - 1/9*a^6 - 1/2*a^5 + 1/3*a^4 + 4/9*a^3 + 1/9*a^2 + 1/18*a + 1/3, 1/918*a^16 + 7/918*a^15 - 11/918*a^14 - 23/306*a^13 - 1/918*a^12 + 20/153*a^11 + 73/459*a^10 - 7/918*a^9 - 35/459*a^8 + 1/27*a^7 - 86/459*a^6 + 421/918*a^5 - 265/918*a^4 - 287/918*a^3 - 397/918*a^2 - 7/17*a + 361/918, 1/892443743838*a^17 + 116991155/297481247946*a^16 + 917158316/148740623973*a^15 + 94295551/3734074242*a^14 - 206168921/3734074242*a^13 - 37246972451/892443743838*a^12 + 34480183991/892443743838*a^11 + 764644733/6466983651*a^10 + 36867991495/297481247946*a^9 - 28510051646/446221871919*a^8 + 355141804543/892443743838*a^7 - 19754763872/446221871919*a^6 - 27112564195/446221871919*a^5 + 157817220161/892443743838*a^4 + 10089029663/38801901906*a^3 + 162334114921/892443743838*a^2 - 162094802527/446221871919*a - 25895071651/892443743838

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

$C_{3}$, which has order $3$

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{3624966722}{26248345407} a^{17} + \frac{699944263}{2916482823} a^{16} + \frac{2844671768}{8749448469} a^{15} - \frac{156287396}{109825713} a^{14} + \frac{84953176}{109825713} a^{13} + \frac{17157904432}{26248345407} a^{12} - \frac{155269516996}{26248345407} a^{11} + \frac{5409210236}{380410803} a^{10} - \frac{19355383378}{2916482823} a^{9} - \frac{1115445049474}{26248345407} a^{8} + \frac{1450174831468}{26248345407} a^{7} + \frac{192424131008}{26248345407} a^{6} - \frac{2307495793820}{26248345407} a^{5} + \frac{664545306056}{26248345407} a^{4} + \frac{13389354656}{1141232409} a^{3} + \frac{185544828760}{26248345407} a^{2} + \frac{32238400738}{26248345407} a + \frac{29430407450}{26248345407} $ -(3624966722)/(26248345407)a^(17) + (699944263)/(2916482823)a^(16) + (2844671768)/(8749448469)a^(15) - (156287396)/(109825713)a^(14) + (84953176)/(109825713)a^(13) + (17157904432)/(26248345407)a^(12) - (155269516996)/(26248345407)a^(11) + (5409210236)/(380410803)a^(10) - (19355383378)/(2916482823)a^(9) - (1115445049474)/(26248345407)a^(8) + (1450174831468)/(26248345407)a^(7) + (192424131008)/(26248345407)a^(6) - (2307495793820)/(26248345407)a^(5) + (664545306056)/(26248345407)a^(4) + (13389354656)/(1141232409)a^(3) + (185544828760)/(26248345407)a^(2) + (32238400738)/(26248345407)*a + (29430407450)/(26248345407) (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:	$\frac{31111425883}{297481247946}a^{17}-\frac{371443856}{2916482823}a^{16}-\frac{26657367895}{99160415982}a^{15}+\frac{541867331}{622345707}a^{14}-\frac{290409677}{1244691414}a^{13}-\frac{31305839968}{148740623973}a^{12}+\frac{620731584688}{148740623973}a^{11}-\frac{18778102490}{2155661217}a^{10}+\frac{113015924900}{49580207991}a^{9}+\frac{4349524158085}{148740623973}a^{8}-\frac{3689362135810}{148740623973}a^{7}-\frac{901636168571}{148740623973}a^{6}+\frac{14642039807275}{297481247946}a^{5}+\frac{745828285495}{148740623973}a^{4}+\frac{225480857939}{12933967302}a^{3}+\frac{39190186937}{148740623973}a^{2}+\frac{188541351343}{297481247946}a-\frac{29712965969}{148740623973}$, $\frac{6918849455}{26248345407}a^{17}-\frac{85654181659}{297481247946}a^{16}-\frac{24447183775}{33053471994}a^{15}+\frac{4007228482}{1867037121}a^{14}-\frac{487295402}{1867037121}a^{13}-\frac{369397615643}{446221871919}a^{12}+\frac{9438017309563}{892443743838}a^{11}-\frac{4920825637}{239517913}a^{10}+\frac{611575354015}{297481247946}a^{9}+\frac{68384344075057}{892443743838}a^{8}-\frac{2851057255631}{52496690814}a^{7}-\frac{13288074594220}{446221871919}a^{6}+\frac{116643054046625}{892443743838}a^{5}+\frac{13045896935069}{446221871919}a^{4}+\frac{618897250412}{19400950953}a^{3}+\frac{4066291942603}{446221871919}a^{2}+\frac{2603676765017}{892443743838}a+\frac{872354640625}{892443743838}$, $\frac{21305852804}{446221871919}a^{17}-\frac{10854863323}{148740623973}a^{16}-\frac{13769443967}{99160415982}a^{15}+\frac{911628239}{1867037121}a^{14}-\frac{548362517}{3734074242}a^{13}-\frac{173724723319}{446221871919}a^{12}+\frac{924863554846}{446221871919}a^{11}-\frac{9566803543}{2155661217}a^{10}+\frac{257763459401}{297481247946}a^{9}+\frac{7250388895114}{446221871919}a^{8}-\frac{7475837532220}{446221871919}a^{7}-\frac{4225126692110}{446221871919}a^{6}+\frac{30801782979607}{892443743838}a^{5}-\frac{982210137887}{446221871919}a^{4}-\frac{488669526487}{38801901906}a^{3}-\frac{229608742690}{446221871919}a^{2}+\frac{249848145770}{446221871919}a+\frac{30761095945}{446221871919}$, $\frac{310675655}{892443743838}a^{17}-\frac{2986038719}{99160415982}a^{16}+\frac{2815742551}{297481247946}a^{15}+\frac{13396292}{109825713}a^{14}-\frac{702070711}{3734074242}a^{13}-\frac{173013738955}{892443743838}a^{12}+\frac{95128758167}{446221871919}a^{11}-\frac{14403150989}{12933967302}a^{10}+\frac{68238079030}{49580207991}a^{9}+\frac{1823414597329}{892443743838}a^{8}-\frac{4282733299715}{446221871919}a^{7}-\frac{716236103573}{892443743838}a^{6}+\frac{10492978501931}{892443743838}a^{5}-\frac{5805646098415}{446221871919}a^{4}-\frac{39267859951}{2282464818}a^{3}-\frac{883892689177}{892443743838}a^{2}+\frac{542183046575}{892443743838}a-\frac{202444508242}{446221871919}$, $\frac{3166393589}{148740623973}a^{17}+\frac{6137374270}{148740623973}a^{16}-\frac{24223623128}{148740623973}a^{15}+\frac{15583264}{622345707}a^{14}+\frac{373621811}{622345707}a^{13}-\frac{58627839065}{148740623973}a^{12}+\frac{97474802914}{148740623973}a^{11}+\frac{6875855485}{6466983651}a^{10}-\frac{918401762578}{148740623973}a^{9}+\frac{26491847584}{2916482823}a^{8}+\frac{713063187223}{49580207991}a^{7}-\frac{421135023757}{16526735997}a^{6}+\frac{465031267349}{49580207991}a^{5}+\frac{652751793553}{16526735997}a^{4}-\frac{5198689774}{718553739}a^{3}+\frac{393599590162}{148740623973}a^{2}-\frac{72336010945}{148740623973}a-\frac{3039031169}{8749448469}$, $\frac{79291655296}{446221871919}a^{17}-\frac{12400506689}{148740623973}a^{16}-\frac{61705862045}{99160415982}a^{15}+\frac{2128545985}{1867037121}a^{14}+\frac{1357897156}{1867037121}a^{13}-\frac{608954726467}{892443743838}a^{12}+\frac{6067482499213}{892443743838}a^{11}-\frac{40692975995}{4311322434}a^{10}-\frac{2181331258235}{297481247946}a^{9}+\frac{23534178028742}{446221871919}a^{8}-\frac{4159307448079}{892443743838}a^{7}-\frac{38547359384219}{892443743838}a^{6}+\frac{67810281219383}{892443743838}a^{5}+\frac{32888220483026}{446221871919}a^{4}+\frac{651848650817}{19400950953}a^{3}+\frac{17428399673195}{892443743838}a^{2}+\frac{3529885218047}{892443743838}a+\frac{1077006016621}{892443743838}$, $\frac{40318581470}{446221871919}a^{17}-\frac{24842294090}{148740623973}a^{16}-\frac{55410508211}{297481247946}a^{15}+\frac{1745257493}{1867037121}a^{14}-\frac{2340045467}{3734074242}a^{13}-\frac{117959652025}{446221871919}a^{12}+\frac{1714718234086}{446221871919}a^{11}-\frac{63256163383}{6466983651}a^{10}+\frac{1716485645155}{297481247946}a^{9}+\frac{11712176945944}{446221871919}a^{8}-\frac{17177134602736}{446221871919}a^{7}+\frac{953916464974}{446221871919}a^{6}+\frac{47653074010273}{892443743838}a^{5}-\frac{10315060065065}{446221871919}a^{4}+\frac{33431468183}{38801901906}a^{3}-\frac{2924653346230}{446221871919}a^{2}-\frac{351679086397}{446221871919}a-\frac{84856150676}{446221871919}$, $\frac{98238587336}{148740623973}a^{17}-\frac{270193589071}{297481247946}a^{16}-\frac{262495256222}{148740623973}a^{15}+\frac{7544395829}{1244691414}a^{14}-\frac{2335348583}{1244691414}a^{13}-\frac{868377293725}{297481247946}a^{12}+\frac{8148555026651}{297481247946}a^{11}-\frac{378511081259}{6466983651}a^{10}+\frac{2240351969948}{148740623973}a^{9}+\frac{195003784390}{972160941}a^{8}-\frac{6383397471967}{33053471994}a^{7}-\frac{3475824940280}{49580207991}a^{6}+\frac{18846057206107}{49580207991}a^{5}-\frac{679116891655}{99160415982}a^{4}-\frac{19323100513}{4311322434}a^{3}-\frac{927478864645}{297481247946}a^{2}+\frac{1045402606171}{297481247946}a-\frac{23510763199}{17498896938}$ 31111425883/297481247946a^17 - 371443856/2916482823a^16 - 26657367895/99160415982a^15 + 541867331/622345707a^14 - 290409677/1244691414a^13 - 31305839968/148740623973a^12 + 620731584688/148740623973a^11 - 18778102490/2155661217a^10 + 113015924900/49580207991a^9 + 4349524158085/148740623973a^8 - 3689362135810/148740623973a^7 - 901636168571/148740623973a^6 + 14642039807275/297481247946a^5 + 745828285495/148740623973a^4 + 225480857939/12933967302a^3 + 39190186937/148740623973a^2 + 188541351343/297481247946a - 29712965969/148740623973, 6918849455/26248345407a^17 - 85654181659/297481247946a^16 - 24447183775/33053471994a^15 + 4007228482/1867037121a^14 - 487295402/1867037121a^13 - 369397615643/446221871919a^12 + 9438017309563/892443743838a^11 - 4920825637/239517913a^10 + 611575354015/297481247946a^9 + 68384344075057/892443743838a^8 - 2851057255631/52496690814a^7 - 13288074594220/446221871919a^6 + 116643054046625/892443743838a^5 + 13045896935069/446221871919a^4 + 618897250412/19400950953a^3 + 4066291942603/446221871919a^2 + 2603676765017/892443743838a + 872354640625/892443743838, 21305852804/446221871919a^17 - 10854863323/148740623973a^16 - 13769443967/99160415982a^15 + 911628239/1867037121a^14 - 548362517/3734074242a^13 - 173724723319/446221871919a^12 + 924863554846/446221871919a^11 - 9566803543/2155661217a^10 + 257763459401/297481247946a^9 + 7250388895114/446221871919a^8 - 7475837532220/446221871919a^7 - 4225126692110/446221871919a^6 + 30801782979607/892443743838a^5 - 982210137887/446221871919a^4 - 488669526487/38801901906a^3 - 229608742690/446221871919a^2 + 249848145770/446221871919a + 30761095945/446221871919, 310675655/892443743838a^17 - 2986038719/99160415982a^16 + 2815742551/297481247946a^15 + 13396292/109825713a^14 - 702070711/3734074242a^13 - 173013738955/892443743838a^12 + 95128758167/446221871919a^11 - 14403150989/12933967302a^10 + 68238079030/49580207991a^9 + 1823414597329/892443743838a^8 - 4282733299715/446221871919a^7 - 716236103573/892443743838a^6 + 10492978501931/892443743838a^5 - 5805646098415/446221871919a^4 - 39267859951/2282464818a^3 - 883892689177/892443743838a^2 + 542183046575/892443743838a - 202444508242/446221871919, 3166393589/148740623973a^17 + 6137374270/148740623973a^16 - 24223623128/148740623973a^15 + 15583264/622345707a^14 + 373621811/622345707a^13 - 58627839065/148740623973a^12 + 97474802914/148740623973a^11 + 6875855485/6466983651a^10 - 918401762578/148740623973a^9 + 26491847584/2916482823a^8 + 713063187223/49580207991a^7 - 421135023757/16526735997a^6 + 465031267349/49580207991a^5 + 652751793553/16526735997a^4 - 5198689774/718553739a^3 + 393599590162/148740623973a^2 - 72336010945/148740623973a - 3039031169/8749448469, 79291655296/446221871919a^17 - 12400506689/148740623973a^16 - 61705862045/99160415982a^15 + 2128545985/1867037121a^14 + 1357897156/1867037121a^13 - 608954726467/892443743838a^12 + 6067482499213/892443743838a^11 - 40692975995/4311322434a^10 - 2181331258235/297481247946a^9 + 23534178028742/446221871919a^8 - 4159307448079/892443743838a^7 - 38547359384219/892443743838a^6 + 67810281219383/892443743838a^5 + 32888220483026/446221871919a^4 + 651848650817/19400950953a^3 + 17428399673195/892443743838a^2 + 3529885218047/892443743838a + 1077006016621/892443743838, 40318581470/446221871919a^17 - 24842294090/148740623973a^16 - 55410508211/297481247946a^15 + 1745257493/1867037121a^14 - 2340045467/3734074242a^13 - 117959652025/446221871919a^12 + 1714718234086/446221871919a^11 - 63256163383/6466983651a^10 + 1716485645155/297481247946a^9 + 11712176945944/446221871919a^8 - 17177134602736/446221871919a^7 + 953916464974/446221871919a^6 + 47653074010273/892443743838a^5 - 10315060065065/446221871919a^4 + 33431468183/38801901906a^3 - 2924653346230/446221871919a^2 - 351679086397/446221871919a - 84856150676/446221871919, 98238587336/148740623973a^17 - 270193589071/297481247946a^16 - 262495256222/148740623973a^15 + 7544395829/1244691414a^14 - 2335348583/1244691414a^13 - 868377293725/297481247946a^12 + 8148555026651/297481247946a^11 - 378511081259/6466983651a^10 + 2240351969948/148740623973a^9 + 195003784390/972160941a^8 - 6383397471967/33053471994a^7 - 3475824940280/49580207991a^6 + 18846057206107/49580207991a^5 - 679116891655/99160415982a^4 - 19323100513/4311322434a^3 - 927478864645/297481247946a^2 + 1045402606171/297481247946a - 23510763199/17498896938	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:	$ 19737.264553893918 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 19737.264553893918 \cdot 3}{6\cdot\sqrt{18283008447003871936512}}\cr\approx \mathstrut & 1.11391474285200 \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 3*x^16 + 8*x^15 - 4*x^13 + 40*x^12 - 74*x^11 - 3*x^10 + 299*x^9 - 181*x^8 - 158*x^7 + 506*x^6 + 166*x^5 + 86*x^4 + 36*x^3 + 9*x^2 + x + 1)

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 - x^17 - 3*x^16 + 8*x^15 - 4*x^13 + 40*x^12 - 74*x^11 - 3*x^10 + 299*x^9 - 181*x^8 - 158*x^7 + 506*x^6 + 166*x^5 + 86*x^4 + 36*x^3 + 9*x^2 + x + 1, 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 - x^17 - 3*x^16 + 8*x^15 - 4*x^13 + 40*x^12 - 74*x^11 - 3*x^10 + 299*x^9 - 181*x^8 - 158*x^7 + 506*x^6 + 166*x^5 + 86*x^4 + 36*x^3 + 9*x^2 + x + 1);

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 - x^17 - 3*x^16 + 8*x^15 - 4*x^13 + 40*x^12 - 74*x^11 - 3*x^10 + 299*x^9 - 181*x^8 - 158*x^7 + 506*x^6 + 166*x^5 + 86*x^4 + 36*x^3 + 9*x^2 + x + 1);

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

$S_3^2$ (as 18T11):

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 36

The 9 conjugacy class representatives for $S_3^2$

Character table for $S_3^2$

Intermediate fields

$\Q(\sqrt{-3}) $, 3.1.1176.1, 3.1.588.1 x3, 6.0.4148928.1, 6.0.1037232.1, 9.1.78066229248.1 x3

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

Galois closure:	data not computed
Degree 6 sibling:	6.2.903168.1
Degree 9 sibling:	9.1.78066229248.1
Degree 12 sibling:	deg 12
Degree 18 siblings:	18.2.780075027072165202624512.1, 18.0.2340225081216495607873536.1
Minimal sibling:	6.2.903168.1

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	${\href{/padicField/5.6.0.1}{6} }^{3}$	R	${\href{/padicField/11.6.0.1}{6} }^{3}$	${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$	${\href{/padicField/17.2.0.1}{2} }^{9}$	${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$	${\href{/padicField/23.2.0.1}{2} }^{9}$	${\href{/padicField/29.6.0.1}{6} }^{3}$	${\href{/padicField/31.3.0.1}{3} }^{6}$	${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$	${\href{/padicField/41.2.0.1}{2} }^{9}$	${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$	${\href{/padicField/47.2.0.1}{2} }^{9}$	${\href{/padicField/53.6.0.1}{6} }^{3}$	${\href{/padicField/59.6.0.1}{6} }^{3}$

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

# 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	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$2$ 2	2.12.22.60	$x^{12} + 14 x^{11} + 85 x^{10} + 314 x^{9} + 832 x^{8} + 1646 x^{7} + 2525 x^{6} + 2970 x^{5} + 2416 x^{4} + 910 x^{3} - 155 x^{2} - 150 x + 25$	$6$	$2$	$22$	$D_6$	$[3]_{3}^{2}$
$3$ 3	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$7$ 7	7.9.6.1	$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$
$7$ 7	7.9.6.1	$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$	$3$	$3$	$6$	$C_3^2$	$[\ ]_{3}^{3}$

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