LMFDB - Number field 18.14.5285331976352154880724992.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 4*x^16 + 9*x^15 - 25*x^14 + 11*x^13 + 105*x^12 + 58*x^11 - 71*x^10 - 273*x^9 - 220*x^8 + 252*x^7 + 374*x^6 + 3*x^5 - 170*x^4 - 64*x^3 + 9*x^2 + 8*x + 1)

gp: K = bnfinit(y^18 - 3*y^17 - 4*y^16 + 9*y^15 - 25*y^14 + 11*y^13 + 105*y^12 + 58*y^11 - 71*y^10 - 273*y^9 - 220*y^8 + 252*y^7 + 374*y^6 + 3*y^5 - 170*y^4 - 64*y^3 + 9*y^2 + 8*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 4*x^16 + 9*x^15 - 25*x^14 + 11*x^13 + 105*x^12 + 58*x^11 - 71*x^10 - 273*x^9 - 220*x^8 + 252*x^7 + 374*x^6 + 3*x^5 - 170*x^4 - 64*x^3 + 9*x^2 + 8*x + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 4*x^16 + 9*x^15 - 25*x^14 + 11*x^13 + 105*x^12 + 58*x^11 - 71*x^10 - 273*x^9 - 220*x^8 + 252*x^7 + 374*x^6 + 3*x^5 - 170*x^4 - 64*x^3 + 9*x^2 + 8*x + 1)

$ x^{18} - 3 x^{17} - 4 x^{16} + 9 x^{15} - 25 x^{14} + 11 x^{13} + 105 x^{12} + 58 x^{11} - 71 x^{10} + \cdots + 1 $ x^18 - 3*x^17 - 4*x^16 + 9*x^15 - 25*x^14 + 11*x^13 + 105*x^12 + 58*x^11 - 71*x^10 - 273*x^9 - 220*x^8 + 252*x^7 + 374*x^6 + 3*x^5 - 170*x^4 - 64*x^3 + 9*x^2 + 8*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:	$[14, 2]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$5285331976352154880724992$ 5285331976352154880724992 $\medspace = 2^{12}\cdot 23^{2}\cdot 37^{9}\cdot 137^{2}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$23.63$	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:	$2^{2/3}23^{1/2}37^{1/2}137^{1/2}\approx 542.0155264272299$
Ramified primes:	$2$, $23$, $37$, $137$ 2, 23, 37, 137	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{37}) $
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{19581}a^{17}+\frac{2624}{19581}a^{16}+\frac{4}{107}a^{15}-\frac{9019}{19581}a^{14}-\frac{6455}{19581}a^{13}-\frac{6655}{19581}a^{12}+\frac{3253}{19581}a^{11}+\frac{1846}{19581}a^{10}+\frac{6337}{19581}a^{9}-\frac{1117}{6527}a^{8}+\frac{542}{6527}a^{7}-\frac{3431}{19581}a^{6}+\frac{308}{6527}a^{5}+\frac{5834}{19581}a^{4}-\frac{6175}{19581}a^{3}+\frac{4333}{19581}a^{2}-\frac{96}{6527}a+\frac{7091}{19581}$ 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, 1/3*a^15 - 1/3*a^14 - 1/3*a^13 - 1/3*a^11 - 1/3*a^10 - 1/3*a^9 - 1/3*a^8 - 1/3*a^7 + 1/3*a^6 + 1/3*a^5 - 1/3*a^3 - 1/3*a^2 - 1/3, 1/3*a^16 + 1/3*a^14 - 1/3*a^13 - 1/3*a^12 + 1/3*a^11 + 1/3*a^10 + 1/3*a^9 + 1/3*a^8 - 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, 1/19581*a^17 + 2624/19581*a^16 + 4/107*a^15 - 9019/19581*a^14 - 6455/19581*a^13 - 6655/19581*a^12 + 3253/19581*a^11 + 1846/19581*a^10 + 6337/19581*a^9 - 1117/6527*a^8 + 542/6527*a^7 - 3431/19581*a^6 + 308/6527*a^5 + 5834/19581*a^4 - 6175/19581*a^3 + 4333/19581*a^2 - 96/6527*a + 7091/19581

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:	$15$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -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:	$\frac{44063867}{19581}a^{17}-\frac{164590171}{19581}a^{16}-\frac{906173}{321}a^{15}+\frac{437384962}{19581}a^{14}-\frac{474419362}{6527}a^{13}+\frac{510290074}{6527}a^{12}+\frac{3502556957}{19581}a^{11}-\frac{21914701}{19581}a^{10}-\frac{3113683288}{19581}a^{9}-\frac{3246512844}{6527}a^{8}-\frac{2531200489}{19581}a^{7}+\frac{12972149399}{19581}a^{6}+\frac{6939831059}{19581}a^{5}-\frac{1659477905}{6527}a^{4}-\frac{3830135933}{19581}a^{3}-\frac{320373}{6527}a^{2}+\frac{397670717}{19581}a+\frac{59877242}{19581}$, $\frac{268761124}{19581}a^{17}-\frac{1012633607}{19581}a^{16}-\frac{4878508}{321}a^{15}+\frac{2647455485}{19581}a^{14}-\frac{2917250789}{6527}a^{13}+\frac{3225196628}{6527}a^{12}+\frac{20792217727}{19581}a^{11}-\frac{377467265}{19581}a^{10}-\frac{18793160933}{19581}a^{9}-\frac{19647355693}{6527}a^{8}-\frac{13871400572}{19581}a^{7}+\frac{78382959307}{19581}a^{6}+\frac{40333612762}{19581}a^{5}-\frac{10055842933}{6527}a^{4}-\frac{22526636986}{19581}a^{3}+\frac{32507167}{6527}a^{2}+\frac{2344152232}{19581}a+\frac{349998601}{19581}$, $\frac{88397255}{19581}a^{17}-\frac{111379263}{6527}a^{16}-\frac{508220}{107}a^{15}+\frac{289410473}{6527}a^{14}-\frac{962375475}{6527}a^{13}+\frac{3223926374}{19581}a^{12}+\frac{6768222460}{19581}a^{11}-\frac{153278528}{19581}a^{10}-\frac{6158335172}{19581}a^{9}-\frac{19328444731}{19581}a^{8}-\frac{1456827150}{6527}a^{7}+\frac{8563457206}{6527}a^{6}+\frac{13022464400}{19581}a^{5}-\frac{9899738020}{19581}a^{4}-\frac{2435643758}{6527}a^{3}+\frac{15118965}{6527}a^{2}+\frac{760993396}{19581}a+\frac{113210444}{19581}$, $\frac{197053681}{19581}a^{17}-\frac{742905872}{19581}a^{16}-\frac{3542530}{321}a^{15}+\frac{1939754123}{19581}a^{14}-\frac{2140013644}{6527}a^{13}+\frac{2370570201}{6527}a^{12}+\frac{15212953984}{19581}a^{11}-\frac{284079110}{19581}a^{10}-\frac{13770875873}{19581}a^{9}-\frac{14397111057}{6527}a^{8}-\frac{10092917960}{19581}a^{7}+\frac{57423869620}{19581}a^{6}+\frac{29478808450}{19581}a^{5}-\frac{7367708114}{6527}a^{4}-\frac{16477825933}{19581}a^{3}+\frac{25144961}{6527}a^{2}+\frac{1714990882}{19581}a+\frac{255942226}{19581}$, $\frac{88397255}{19581}a^{17}-\frac{111379263}{6527}a^{16}-\frac{508220}{107}a^{15}+\frac{289410473}{6527}a^{14}-\frac{962375475}{6527}a^{13}+\frac{3223926374}{19581}a^{12}+\frac{6768222460}{19581}a^{11}-\frac{153278528}{19581}a^{10}-\frac{6158335172}{19581}a^{9}-\frac{19328444731}{19581}a^{8}-\frac{1456827150}{6527}a^{7}+\frac{8563457206}{6527}a^{6}+\frac{13022464400}{19581}a^{5}-\frac{9899738020}{19581}a^{4}-\frac{2435643758}{6527}a^{3}+\frac{15118965}{6527}a^{2}+\frac{760973815}{19581}a+\frac{113210444}{19581}$, $\frac{108656426}{19581}a^{17}-\frac{408768083}{19581}a^{16}-\frac{2017870}{321}a^{15}+\frac{1071522704}{19581}a^{14}-\frac{1177638169}{6527}a^{13}+\frac{3887784229}{19581}a^{12}+\frac{2814910508}{6527}a^{11}-\frac{43600194}{6527}a^{10}-\frac{2537513567}{6527}a^{9}-\frac{23862888440}{19581}a^{8}-\frac{5722436510}{19581}a^{7}+\frac{31733498002}{19581}a^{6}+\frac{16456344050}{19581}a^{5}-\frac{12203386322}{19581}a^{4}-\frac{9170894659}{19581}a^{3}+\frac{10025996}{6527}a^{2}+\frac{317999162}{6527}a+\frac{142731782}{19581}$, $a$, $\frac{194019346}{19581}a^{17}-\frac{244027043}{6527}a^{16}-\frac{1147504}{107}a^{15}+\frac{636167844}{6527}a^{14}-\frac{2108738561}{6527}a^{13}+\frac{7026003784}{19581}a^{12}+\frac{14938864640}{19581}a^{11}-\frac{297696820}{19581}a^{10}-\frac{13544329963}{19581}a^{9}-\frac{42494593232}{19581}a^{8}-\frac{3275570626}{6527}a^{7}+\frac{18829646342}{6527}a^{6}+\frac{28883858872}{19581}a^{5}-\frac{21748215098}{19581}a^{4}-\frac{5388570811}{6527}a^{3}+\frac{26922432}{6527}a^{2}+\frac{1682977451}{19581}a+\frac{251022274}{19581}$, $\frac{29963483}{19581}a^{17}-\frac{38114617}{6527}a^{16}-\frac{145391}{107}a^{15}+\frac{97234178}{6527}a^{14}-\frac{329106081}{6527}a^{13}+\frac{1134782339}{19581}a^{12}+\frac{2222770996}{19581}a^{11}-\frac{80760509}{19581}a^{10}-\frac{2063390543}{19581}a^{9}-\frac{6494231437}{19581}a^{8}-\frac{429365121}{6527}a^{7}+\frac{2871481109}{6527}a^{6}+\frac{4169522981}{19581}a^{5}-\frac{3329412334}{19581}a^{4}-\frac{791571623}{6527}a^{3}+\frac{9197728}{6527}a^{2}+\frac{248026663}{19581}a+\frac{36527873}{19581}$, $\frac{283630226}{19581}a^{17}-\frac{1069825757}{19581}a^{16}-\frac{5059783}{321}a^{15}+\frac{2790608573}{19581}a^{14}-\frac{3081586345}{6527}a^{13}+\frac{10256664622}{19581}a^{12}+\frac{7287113532}{6527}a^{11}-\frac{140108798}{6527}a^{10}-\frac{6603236943}{6527}a^{9}-\frac{62141726426}{19581}a^{8}-\frac{14435558717}{19581}a^{7}+\frac{82603824688}{19581}a^{6}+\frac{42318769613}{19581}a^{5}-\frac{31795442750}{19581}a^{4}-\frac{23672607556}{19581}a^{3}+\frac{37282014}{6527}a^{2}+\frac{821437452}{6527}a+\frac{367716146}{19581}$, $\frac{19395784}{6527}a^{17}-\frac{72788941}{6527}a^{16}-\frac{1118863}{321}a^{15}+\frac{574602319}{19581}a^{14}-\frac{1887108494}{19581}a^{13}+\frac{687222887}{6527}a^{12}+\frac{4553573770}{19581}a^{11}-\frac{46996031}{19581}a^{10}-\frac{4091307461}{19581}a^{9}-\frac{12807606173}{19581}a^{8}-\frac{3165173999}{19581}a^{7}+\frac{17025923546}{19581}a^{6}+\frac{8950222142}{19581}a^{5}-\frac{2178796218}{6527}a^{4}-\frac{4970499797}{19581}a^{3}+\frac{6394693}{19581}a^{2}+\frac{172244567}{6527}a+\frac{77620990}{19581}$, $\frac{11335570}{19581}a^{17}-\frac{14793008}{6527}a^{16}-\frac{27432}{107}a^{15}+\frac{35964444}{6527}a^{14}-\frac{127574949}{6527}a^{13}+\frac{472566994}{19581}a^{12}+\frac{768805163}{19581}a^{11}-\frac{64221028}{19581}a^{10}-\frac{754715431}{19581}a^{9}-\frac{2397934823}{19581}a^{8}-\frac{94993172}{6527}a^{7}+\frac{1056282766}{6527}a^{6}+\frac{1318160782}{19581}a^{5}-\frac{1236982727}{19581}a^{4}-\frac{263322324}{6527}a^{3}+\frac{8019081}{6527}a^{2}+\frac{83282585}{19581}a+\frac{11865796}{19581}$, $\frac{146198411}{6527}a^{17}-\frac{1652914207}{19581}a^{16}-\frac{2644454}{107}a^{15}+\frac{4319659556}{19581}a^{14}-\frac{14285295269}{19581}a^{13}+\frac{15804313849}{19581}a^{12}+\frac{33907245020}{19581}a^{11}-\frac{628640893}{19581}a^{10}-\frac{30660706849}{19581}a^{9}-\frac{96166941814}{19581}a^{8}-\frac{7522469462}{6527}a^{7}+\frac{127886920546}{19581}a^{6}+\frac{65729987768}{19581}a^{5}-\frac{49228258937}{19581}a^{4}-\frac{36722642575}{19581}a^{3}+\frac{165954439}{19581}a^{2}+\frac{3821674240}{19581}a+\frac{570347365}{19581}$, $\frac{309417274}{19581}a^{17}-\frac{389149778}{6527}a^{16}-\frac{1831637}{107}a^{15}+\frac{1014662348}{6527}a^{14}-\frac{3362874925}{6527}a^{13}+\frac{11202427840}{19581}a^{12}+\frac{23829650888}{19581}a^{11}-\frac{476845708}{19581}a^{10}-\frac{21601516804}{19581}a^{9}-\frac{67771083551}{19581}a^{8}-\frac{5226411515}{6527}a^{7}+\frac{30033136156}{6527}a^{6}+\frac{46069413382}{19581}a^{5}-\frac{34693431593}{19581}a^{4}-\frac{8594016122}{6527}a^{3}+\frac{43807936}{6527}a^{2}+\frac{2684029559}{19581}a+\frac{400138645}{19581}$, $\frac{125112758}{19581}a^{17}-\frac{471332188}{19581}a^{16}-\frac{2275960}{321}a^{15}+\frac{410867785}{6527}a^{14}-\frac{4073549377}{19581}a^{13}+\frac{1500517733}{6527}a^{12}+\frac{9683566963}{19581}a^{11}-\frac{174047267}{19581}a^{10}-\frac{8750447300}{19581}a^{9}-\frac{27441656647}{19581}a^{8}-\frac{6468931490}{19581}a^{7}+\frac{12164986299}{6527}a^{6}+\frac{6263635822}{6527}a^{5}-\frac{4682221978}{6527}a^{4}-\frac{3497545684}{6527}a^{3}+\frac{45492863}{19581}a^{2}+\frac{1091776715}{19581}a+\frac{162923149}{19581}$ 44063867/19581a^17 - 164590171/19581a^16 - 906173/321a^15 + 437384962/19581a^14 - 474419362/6527a^13 + 510290074/6527a^12 + 3502556957/19581a^11 - 21914701/19581a^10 - 3113683288/19581a^9 - 3246512844/6527a^8 - 2531200489/19581a^7 + 12972149399/19581a^6 + 6939831059/19581a^5 - 1659477905/6527a^4 - 3830135933/19581a^3 - 320373/6527a^2 + 397670717/19581a + 59877242/19581, 268761124/19581a^17 - 1012633607/19581a^16 - 4878508/321a^15 + 2647455485/19581a^14 - 2917250789/6527a^13 + 3225196628/6527a^12 + 20792217727/19581a^11 - 377467265/19581a^10 - 18793160933/19581a^9 - 19647355693/6527a^8 - 13871400572/19581a^7 + 78382959307/19581a^6 + 40333612762/19581a^5 - 10055842933/6527a^4 - 22526636986/19581a^3 + 32507167/6527a^2 + 2344152232/19581a + 349998601/19581, 88397255/19581a^17 - 111379263/6527a^16 - 508220/107a^15 + 289410473/6527a^14 - 962375475/6527a^13 + 3223926374/19581a^12 + 6768222460/19581a^11 - 153278528/19581a^10 - 6158335172/19581a^9 - 19328444731/19581a^8 - 1456827150/6527a^7 + 8563457206/6527a^6 + 13022464400/19581a^5 - 9899738020/19581a^4 - 2435643758/6527a^3 + 15118965/6527a^2 + 760993396/19581a + 113210444/19581, 197053681/19581a^17 - 742905872/19581a^16 - 3542530/321a^15 + 1939754123/19581a^14 - 2140013644/6527a^13 + 2370570201/6527a^12 + 15212953984/19581a^11 - 284079110/19581a^10 - 13770875873/19581a^9 - 14397111057/6527a^8 - 10092917960/19581a^7 + 57423869620/19581a^6 + 29478808450/19581a^5 - 7367708114/6527a^4 - 16477825933/19581a^3 + 25144961/6527a^2 + 1714990882/19581a + 255942226/19581, 88397255/19581a^17 - 111379263/6527a^16 - 508220/107a^15 + 289410473/6527a^14 - 962375475/6527a^13 + 3223926374/19581a^12 + 6768222460/19581a^11 - 153278528/19581a^10 - 6158335172/19581a^9 - 19328444731/19581a^8 - 1456827150/6527a^7 + 8563457206/6527a^6 + 13022464400/19581a^5 - 9899738020/19581a^4 - 2435643758/6527a^3 + 15118965/6527a^2 + 760973815/19581a + 113210444/19581, 108656426/19581a^17 - 408768083/19581a^16 - 2017870/321a^15 + 1071522704/19581a^14 - 1177638169/6527a^13 + 3887784229/19581a^12 + 2814910508/6527a^11 - 43600194/6527a^10 - 2537513567/6527a^9 - 23862888440/19581a^8 - 5722436510/19581a^7 + 31733498002/19581a^6 + 16456344050/19581a^5 - 12203386322/19581a^4 - 9170894659/19581a^3 + 10025996/6527a^2 + 317999162/6527a + 142731782/19581, a, 194019346/19581a^17 - 244027043/6527a^16 - 1147504/107a^15 + 636167844/6527a^14 - 2108738561/6527a^13 + 7026003784/19581a^12 + 14938864640/19581a^11 - 297696820/19581a^10 - 13544329963/19581a^9 - 42494593232/19581a^8 - 3275570626/6527a^7 + 18829646342/6527a^6 + 28883858872/19581a^5 - 21748215098/19581a^4 - 5388570811/6527a^3 + 26922432/6527a^2 + 1682977451/19581a + 251022274/19581, 29963483/19581a^17 - 38114617/6527a^16 - 145391/107a^15 + 97234178/6527a^14 - 329106081/6527a^13 + 1134782339/19581a^12 + 2222770996/19581a^11 - 80760509/19581a^10 - 2063390543/19581a^9 - 6494231437/19581a^8 - 429365121/6527a^7 + 2871481109/6527a^6 + 4169522981/19581a^5 - 3329412334/19581a^4 - 791571623/6527a^3 + 9197728/6527a^2 + 248026663/19581a + 36527873/19581, 283630226/19581a^17 - 1069825757/19581a^16 - 5059783/321a^15 + 2790608573/19581a^14 - 3081586345/6527a^13 + 10256664622/19581a^12 + 7287113532/6527a^11 - 140108798/6527a^10 - 6603236943/6527a^9 - 62141726426/19581a^8 - 14435558717/19581a^7 + 82603824688/19581a^6 + 42318769613/19581a^5 - 31795442750/19581a^4 - 23672607556/19581a^3 + 37282014/6527a^2 + 821437452/6527a + 367716146/19581, 19395784/6527a^17 - 72788941/6527a^16 - 1118863/321a^15 + 574602319/19581a^14 - 1887108494/19581a^13 + 687222887/6527a^12 + 4553573770/19581a^11 - 46996031/19581a^10 - 4091307461/19581a^9 - 12807606173/19581a^8 - 3165173999/19581a^7 + 17025923546/19581a^6 + 8950222142/19581a^5 - 2178796218/6527a^4 - 4970499797/19581a^3 + 6394693/19581a^2 + 172244567/6527a + 77620990/19581, 11335570/19581a^17 - 14793008/6527a^16 - 27432/107a^15 + 35964444/6527a^14 - 127574949/6527a^13 + 472566994/19581a^12 + 768805163/19581a^11 - 64221028/19581a^10 - 754715431/19581a^9 - 2397934823/19581a^8 - 94993172/6527a^7 + 1056282766/6527a^6 + 1318160782/19581a^5 - 1236982727/19581a^4 - 263322324/6527a^3 + 8019081/6527a^2 + 83282585/19581a + 11865796/19581, 146198411/6527a^17 - 1652914207/19581a^16 - 2644454/107a^15 + 4319659556/19581a^14 - 14285295269/19581a^13 + 15804313849/19581a^12 + 33907245020/19581a^11 - 628640893/19581a^10 - 30660706849/19581a^9 - 96166941814/19581a^8 - 7522469462/6527a^7 + 127886920546/19581a^6 + 65729987768/19581a^5 - 49228258937/19581a^4 - 36722642575/19581a^3 + 165954439/19581a^2 + 3821674240/19581a + 570347365/19581, 309417274/19581a^17 - 389149778/6527a^16 - 1831637/107a^15 + 1014662348/6527a^14 - 3362874925/6527a^13 + 11202427840/19581a^12 + 23829650888/19581a^11 - 476845708/19581a^10 - 21601516804/19581a^9 - 67771083551/19581a^8 - 5226411515/6527a^7 + 30033136156/6527a^6 + 46069413382/19581a^5 - 34693431593/19581a^4 - 8594016122/6527a^3 + 43807936/6527a^2 + 2684029559/19581a + 400138645/19581, 125112758/19581a^17 - 471332188/19581a^16 - 2275960/321a^15 + 410867785/6527a^14 - 4073549377/19581a^13 + 1500517733/6527a^12 + 9683566963/19581a^11 - 174047267/19581a^10 - 8750447300/19581a^9 - 27441656647/19581a^8 - 6468931490/19581a^7 + 12164986299/6527a^6 + 6263635822/6527a^5 - 4682221978/6527a^4 - 3497545684/6527a^3 + 45492863/19581a^2 + 1091776715/19581a + 162923149/19581 (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:	$ 1067608.94817 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{2}\cdot 1067608.94817 \cdot 1}{2\cdot\sqrt{5285331976352154880724992}}\cr\approx \mathstrut & 0.150184720298 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 4*x^16 + 9*x^15 - 25*x^14 + 11*x^13 + 105*x^12 + 58*x^11 - 71*x^10 - 273*x^9 - 220*x^8 + 252*x^7 + 374*x^6 + 3*x^5 - 170*x^4 - 64*x^3 + 9*x^2 + 8*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 - 3*x^17 - 4*x^16 + 9*x^15 - 25*x^14 + 11*x^13 + 105*x^12 + 58*x^11 - 71*x^10 - 273*x^9 - 220*x^8 + 252*x^7 + 374*x^6 + 3*x^5 - 170*x^4 - 64*x^3 + 9*x^2 + 8*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 - 3*x^17 - 4*x^16 + 9*x^15 - 25*x^14 + 11*x^13 + 105*x^12 + 58*x^11 - 71*x^10 - 273*x^9 - 220*x^8 + 252*x^7 + 374*x^6 + 3*x^5 - 170*x^4 - 64*x^3 + 9*x^2 + 8*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 - 3*x^17 - 4*x^16 + 9*x^15 - 25*x^14 + 11*x^13 + 105*x^12 + 58*x^11 - 71*x^10 - 273*x^9 - 220*x^8 + 252*x^7 + 374*x^6 + 3*x^5 - 170*x^4 - 64*x^3 + 9*x^2 + 8*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

$S_3\wr S_3$ (as 18T319):

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 1296

The 22 conjugacy class representatives for $S_3\wr S_3$

Character table for $S_3\wr S_3$

Intermediate fields

$\Q(\sqrt{37}) $, 3.3.148.1 x3, 6.6.810448.1, 9.7.10214886592.1

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:	data not computed
Degree 12 sibling:	data not computed
Degree 18 siblings:	data not computed
Degree 24 siblings:	data not computed
Degree 27 siblings:	data not computed
Degree 36 siblings:	data not computed
Minimal sibling:	9.7.10214886592.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$	${\href{/padicField/5.6.0.1}{6} }^{3}$	${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$	${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$	${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$	${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$	${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$	R	${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$	${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$	R	${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$	${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$	${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$	${\href{/padicField/53.9.0.1}{9} }^{2}$	${\href{/padicField/59.6.0.1}{6} }^{3}$

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	2.18.12.1	$x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$	$3$	$6$	$12$	$S_3 \times C_3$	$[\ ]_{3}^{6}$
$23$ 23	23.2.0.1	$x^{2} + 21 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.6.0.1	$x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	23.6.0.1	$x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$37$ 37	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} + 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.4.2.1	$x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	37.4.2.1	$x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	37.4.2.1	$x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$137$ 137	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{137}$	$x + 134$	$1$	$1$	$0$	Trivial	$[\ ]$
	137.2.1.1	$x^{2} + 137$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	137.2.0.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	137.2.1.1	$x^{2} + 137$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	137.2.0.1	$x^{2} + 131 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	137.3.0.1	$x^{3} + 6 x + 134$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	137.3.0.1	$x^{3} + 6 x + 134$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$

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