Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 25*x^16 + 20*x^15 + 232*x^14 - 137*x^13 - 1018*x^12 + 403*x^11 + 2291*x^10 - 566*x^9 - 2742*x^8 + 387*x^7 + 1711*x^6 - 131*x^5 - 505*x^4 + 30*x^3 + 55*x^2 - 5*x - 1)
gp: K = bnfinit(y^18 - y^17 - 25*y^16 + 20*y^15 + 232*y^14 - 137*y^13 - 1018*y^12 + 403*y^11 + 2291*y^10 - 566*y^9 - 2742*y^8 + 387*y^7 + 1711*y^6 - 131*y^5 - 505*y^4 + 30*y^3 + 55*y^2 - 5*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 - 25*x^16 + 20*x^15 + 232*x^14 - 137*x^13 - 1018*x^12 + 403*x^11 + 2291*x^10 - 566*x^9 - 2742*x^8 + 387*x^7 + 1711*x^6 - 131*x^5 - 505*x^4 + 30*x^3 + 55*x^2 - 5*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 - 25*x^16 + 20*x^15 + 232*x^14 - 137*x^13 - 1018*x^12 + 403*x^11 + 2291*x^10 - 566*x^9 - 2742*x^8 + 387*x^7 + 1711*x^6 - 131*x^5 - 505*x^4 + 30*x^3 + 55*x^2 - 5*x - 1)
\( x^{18} - x^{17} - 25 x^{16} + 20 x^{15} + 232 x^{14} - 137 x^{13} - 1018 x^{12} + 403 x^{11} + 2291 x^{10} + \cdots - 1 \)
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: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(563362135874260093126953125\)
\(\medspace = 5^{9}\cdot 19^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.63\) | 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: | $5^{1/2}19^{8/9}\approx 30.630555264593426$ | ||
| Ramified primes: |
\(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(95=5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{95}(64,·)$, $\chi_{95}(1,·)$, $\chi_{95}(66,·)$, $\chi_{95}(4,·)$, $\chi_{95}(6,·)$, $\chi_{95}(9,·)$, $\chi_{95}(74,·)$, $\chi_{95}(11,·)$, $\chi_{95}(16,·)$, $\chi_{95}(81,·)$, $\chi_{95}(24,·)$, $\chi_{95}(26,·)$, $\chi_{95}(36,·)$, $\chi_{95}(39,·)$, $\chi_{95}(44,·)$, $\chi_{95}(49,·)$, $\chi_{95}(54,·)$, $\chi_{95}(61,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{4130024167189}a^{17}+\frac{920978222672}{4130024167189}a^{16}+\frac{606794707934}{4130024167189}a^{15}-\frac{492036479265}{4130024167189}a^{14}+\frac{629306648510}{4130024167189}a^{13}-\frac{1857958233227}{4130024167189}a^{12}-\frac{550922843041}{4130024167189}a^{11}-\frac{1554275344905}{4130024167189}a^{10}-\frac{1498768582140}{4130024167189}a^{9}+\frac{657444201028}{4130024167189}a^{8}+\frac{846780531398}{4130024167189}a^{7}-\frac{1207865625684}{4130024167189}a^{6}+\frac{446543373733}{4130024167189}a^{5}+\frac{491195884715}{4130024167189}a^{4}+\frac{670693194266}{4130024167189}a^{3}+\frac{1281532435586}{4130024167189}a^{2}+\frac{20048467778}{4130024167189}a+\frac{983742514991}{4130024167189}$
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 $2$)
| 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: |
$\frac{1074519814563}{4130024167189}a^{17}+\frac{3428587421918}{4130024167189}a^{16}-\frac{33593120081742}{4130024167189}a^{15}-\frac{86832424797955}{4130024167189}a^{14}+\frac{391318433084339}{4130024167189}a^{13}+\frac{805102378422971}{4130024167189}a^{12}-\frac{21\!\cdots\!55}{4130024167189}a^{11}-\frac{34\!\cdots\!89}{4130024167189}a^{10}+\frac{59\!\cdots\!64}{4130024167189}a^{9}+\frac{72\!\cdots\!13}{4130024167189}a^{8}-\frac{84\!\cdots\!02}{4130024167189}a^{7}-\frac{75\!\cdots\!11}{4130024167189}a^{6}+\frac{60\!\cdots\!38}{4130024167189}a^{5}+\frac{37\!\cdots\!28}{4130024167189}a^{4}-\frac{19\!\cdots\!20}{4130024167189}a^{3}-\frac{726553844862815}{4130024167189}a^{2}+\frac{186143213382771}{4130024167189}a+\frac{17485591039761}{4130024167189}$, $\frac{622570799780}{4130024167189}a^{17}+\frac{1789324326870}{4130024167189}a^{16}-\frac{19317911272545}{4130024167189}a^{15}-\frac{45268953032230}{4130024167189}a^{14}+\frac{223784680259792}{4130024167189}a^{13}+\frac{418926859039860}{4130024167189}a^{12}-\frac{12\!\cdots\!64}{4130024167189}a^{11}-\frac{17\!\cdots\!00}{4130024167189}a^{10}+\frac{33\!\cdots\!30}{4130024167189}a^{9}+\frac{37\!\cdots\!70}{4130024167189}a^{8}-\frac{47\!\cdots\!98}{4130024167189}a^{7}-\frac{40\!\cdots\!47}{4130024167189}a^{6}+\frac{34\!\cdots\!74}{4130024167189}a^{5}+\frac{21\!\cdots\!38}{4130024167189}a^{4}-\frac{10\!\cdots\!08}{4130024167189}a^{3}-\frac{489294784074778}{4130024167189}a^{2}+\frac{102340541668796}{4130024167189}a+\frac{21464603368731}{4130024167189}$, $\frac{4066037481795}{4130024167189}a^{17}-\frac{2630914697637}{4130024167189}a^{16}-\frac{102457565160157}{4130024167189}a^{15}+\frac{45614841984178}{4130024167189}a^{14}+\frac{955584448567583}{4130024167189}a^{13}-\frac{230864770934681}{4130024167189}a^{12}-\frac{41\!\cdots\!01}{4130024167189}a^{11}+\frac{260842847290735}{4130024167189}a^{10}+\frac{91\!\cdots\!70}{4130024167189}a^{9}+\frac{568908131717448}{4130024167189}a^{8}-\frac{10\!\cdots\!83}{4130024167189}a^{7}-\frac{14\!\cdots\!28}{4130024167189}a^{6}+\frac{58\!\cdots\!75}{4130024167189}a^{5}+\frac{10\!\cdots\!65}{4130024167189}a^{4}-\frac{13\!\cdots\!59}{4130024167189}a^{3}-\frac{257352650937903}{4130024167189}a^{2}+\frac{86926398269413}{4130024167189}a+\frac{13795559887304}{4130024167189}$, $\frac{3309187281925}{4130024167189}a^{17}-\frac{1761545616300}{4130024167189}a^{16}-\frac{83966087673525}{4130024167189}a^{15}+\frac{27852230159430}{4130024167189}a^{14}+\frac{790644848743652}{4130024167189}a^{13}-\frac{104561920300352}{4130024167189}a^{12}-\frac{35\!\cdots\!40}{4130024167189}a^{11}-\frac{138196191767620}{4130024167189}a^{10}+\frac{78\!\cdots\!34}{4130024167189}a^{9}+\frac{12\!\cdots\!06}{4130024167189}a^{8}-\frac{93\!\cdots\!26}{4130024167189}a^{7}-\frac{20\!\cdots\!54}{4130024167189}a^{6}+\frac{55\!\cdots\!42}{4130024167189}a^{5}+\frac{14\!\cdots\!13}{4130024167189}a^{4}-\frac{14\!\cdots\!29}{4130024167189}a^{3}-\frac{370401057067681}{4130024167189}a^{2}+\frac{106013925874152}{4130024167189}a+\frac{18158893443258}{4130024167189}$, $\frac{9453637026195}{4130024167189}a^{17}-\frac{12774882845907}{4130024167189}a^{16}-\frac{229230590824105}{4130024167189}a^{15}+\frac{265668219940913}{4130024167189}a^{14}+\frac{20\!\cdots\!10}{4130024167189}a^{13}-\frac{19\!\cdots\!43}{4130024167189}a^{12}-\frac{83\!\cdots\!50}{4130024167189}a^{11}+\frac{61\!\cdots\!61}{4130024167189}a^{10}+\frac{17\!\cdots\!45}{4130024167189}a^{9}-\frac{93\!\cdots\!40}{4130024167189}a^{8}-\frac{18\!\cdots\!20}{4130024167189}a^{7}+\frac{67\!\cdots\!31}{4130024167189}a^{6}+\frac{93\!\cdots\!75}{4130024167189}a^{5}-\frac{19\!\cdots\!07}{4130024167189}a^{4}-\frac{19\!\cdots\!28}{4130024167189}a^{3}+\frac{88981835821022}{4130024167189}a^{2}+\frac{81838530676626}{4130024167189}a+\frac{7677594406061}{4130024167189}$, $\frac{980591323975}{4130024167189}a^{17}-\frac{3115316491620}{4130024167189}a^{16}-\frac{20511812711280}{4130024167189}a^{15}+\frac{69417492999000}{4130024167189}a^{14}+\frac{141328758729620}{4130024167189}a^{13}-\frac{552600275902280}{4130024167189}a^{12}-\frac{342375512350314}{4130024167189}a^{11}+\frac{19\!\cdots\!40}{4130024167189}a^{10}+\frac{22951215010811}{4130024167189}a^{9}-\frac{34\!\cdots\!54}{4130024167189}a^{8}+\frac{941390164364744}{4130024167189}a^{7}+\frac{28\!\cdots\!83}{4130024167189}a^{6}-\frac{11\!\cdots\!86}{4130024167189}a^{5}-\frac{999040434798075}{4130024167189}a^{4}+\frac{417535719993515}{4130024167189}a^{3}+\frac{100892432130971}{4130024167189}a^{2}-\frac{24466710753604}{4130024167189}a+\frac{188435274677}{4130024167189}$, $\frac{3022678010986}{4130024167189}a^{17}-\frac{2247336136760}{4130024167189}a^{16}-\frac{76429482069035}{4130024167189}a^{15}+\frac{40839517217965}{4130024167189}a^{14}+\frac{718980732398917}{4130024167189}a^{13}-\frac{228504998920433}{4130024167189}a^{12}-\frac{32\!\cdots\!03}{4130024167189}a^{11}+\frac{379231575522380}{4130024167189}a^{10}+\frac{72\!\cdots\!51}{4130024167189}a^{9}+\frac{244039787039934}{4130024167189}a^{8}-\frac{87\!\cdots\!97}{4130024167189}a^{7}-\frac{12\!\cdots\!66}{4130024167189}a^{6}+\frac{52\!\cdots\!44}{4130024167189}a^{5}+\frac{10\!\cdots\!32}{4130024167189}a^{4}-\frac{13\!\cdots\!62}{4130024167189}a^{3}-\frac{285939887038671}{4130024167189}a^{2}+\frac{89772040603365}{4130024167189}a+\frac{12640182733562}{4130024167189}$, $\frac{7973207800700}{4130024167189}a^{17}-\frac{8292094265000}{4130024167189}a^{16}-\frac{197337251803925}{4130024167189}a^{15}+\frac{164477901053155}{4130024167189}a^{14}+\frac{18\!\cdots\!25}{4130024167189}a^{13}-\frac{11\!\cdots\!55}{4130024167189}a^{12}-\frac{77\!\cdots\!75}{4130024167189}a^{11}+\frac{30\!\cdots\!10}{4130024167189}a^{10}+\frac{16\!\cdots\!25}{4130024167189}a^{9}-\frac{36\!\cdots\!50}{4130024167189}a^{8}-\frac{18\!\cdots\!25}{4130024167189}a^{7}+\frac{12\!\cdots\!20}{4130024167189}a^{6}+\frac{10\!\cdots\!38}{4130024167189}a^{5}+\frac{513390576903330}{4130024167189}a^{4}-\frac{25\!\cdots\!85}{4130024167189}a^{3}-\frac{324002674799455}{4130024167189}a^{2}+\frac{148752381167415}{4130024167189}a+\frac{15113233157871}{4130024167189}$, $\frac{8953799124675}{4130024167189}a^{17}-\frac{11407410756620}{4130024167189}a^{16}-\frac{217849064515205}{4130024167189}a^{15}+\frac{233895394052155}{4130024167189}a^{14}+\frac{19\!\cdots\!45}{4130024167189}a^{13}-\frac{16\!\cdots\!35}{4130024167189}a^{12}-\frac{80\!\cdots\!89}{4130024167189}a^{11}+\frac{50\!\cdots\!50}{4130024167189}a^{10}+\frac{16\!\cdots\!36}{4130024167189}a^{9}-\frac{70\!\cdots\!04}{4130024167189}a^{8}-\frac{17\!\cdots\!81}{4130024167189}a^{7}+\frac{41\!\cdots\!03}{4130024167189}a^{6}+\frac{94\!\cdots\!52}{4130024167189}a^{5}-\frac{485649857894745}{4130024167189}a^{4}-\frac{21\!\cdots\!70}{4130024167189}a^{3}-\frac{223110242668484}{4130024167189}a^{2}+\frac{124285670413811}{4130024167189}a+\frac{23561716766926}{4130024167189}$, $\frac{6358233562675}{4130024167189}a^{17}-\frac{5657446633925}{4130024167189}a^{16}-\frac{157860712358393}{4130024167189}a^{15}+\frac{108203633762930}{4130024167189}a^{14}+\frac{14\!\cdots\!32}{4130024167189}a^{13}-\frac{681812330518852}{4130024167189}a^{12}-\frac{61\!\cdots\!31}{4130024167189}a^{11}+\frac{16\!\cdots\!50}{4130024167189}a^{10}+\frac{13\!\cdots\!24}{4130024167189}a^{9}-\frac{16\!\cdots\!14}{4130024167189}a^{8}-\frac{14\!\cdots\!11}{4130024167189}a^{7}+\frac{101702064419811}{4130024167189}a^{6}+\frac{80\!\cdots\!38}{4130024167189}a^{5}+\frac{631346744382473}{4130024167189}a^{4}-\frac{18\!\cdots\!74}{4130024167189}a^{3}-\frac{235262309940661}{4130024167189}a^{2}+\frac{112628620091765}{4130024167189}a+\frac{8713690039003}{4130024167189}$, $\frac{658774140521}{4130024167189}a^{17}+\frac{156875710377}{4130024167189}a^{16}-\frac{16779240179210}{4130024167189}a^{15}-\frac{7089460996468}{4130024167189}a^{14}+\frac{156881313180662}{4130024167189}a^{13}+\frac{92776157347248}{4130024167189}a^{12}-\frac{676473654237434}{4130024167189}a^{11}-\frac{480791137808236}{4130024167189}a^{10}+\frac{14\!\cdots\!70}{4130024167189}a^{9}+\frac{10\!\cdots\!50}{4130024167189}a^{8}-\frac{16\!\cdots\!00}{4130024167189}a^{7}-\frac{999255372731559}{4130024167189}a^{6}+\frac{950230357865034}{4130024167189}a^{5}+\frac{364144002285629}{4130024167189}a^{4}-\frac{235531960016133}{4130024167189}a^{3}-\frac{27182290455101}{4130024167189}a^{2}+\frac{2045316040427}{4130024167189}a-\frac{1078692444287}{4130024167189}$, $\frac{8379117211632}{4130024167189}a^{17}-\frac{16203470267825}{4130024167189}a^{16}-\frac{195637470742363}{4130024167189}a^{15}+\frac{352500644738868}{4130024167189}a^{14}+\frac{16\!\cdots\!71}{4130024167189}a^{13}-\frac{27\!\cdots\!14}{4130024167189}a^{12}-\frac{62\!\cdots\!95}{4130024167189}a^{11}+\frac{95\!\cdots\!50}{4130024167189}a^{10}+\frac{11\!\cdots\!81}{4130024167189}a^{9}-\frac{16\!\cdots\!53}{4130024167189}a^{8}-\frac{97\!\cdots\!18}{4130024167189}a^{7}+\frac{14\!\cdots\!42}{4130024167189}a^{6}+\frac{32\!\cdots\!37}{4130024167189}a^{5}-\frac{57\!\cdots\!35}{4130024167189}a^{4}-\frac{14005118126208}{4130024167189}a^{3}+\frac{815535680683837}{4130024167189}a^{2}-\frac{104304682706145}{4130024167189}a-\frac{9807996633700}{4130024167189}$, $\frac{1152497666623}{4130024167189}a^{17}-\frac{7683447244728}{4130024167189}a^{16}-\frac{18374571294361}{4130024167189}a^{15}+\frac{179241710643445}{4130024167189}a^{14}+\frac{45138978362685}{4130024167189}a^{13}-\frac{15\!\cdots\!51}{4130024167189}a^{12}+\frac{496405414910167}{4130024167189}a^{11}+\frac{59\!\cdots\!89}{4130024167189}a^{10}-\frac{29\!\cdots\!54}{4130024167189}a^{9}-\frac{11\!\cdots\!23}{4130024167189}a^{8}+\frac{59\!\cdots\!04}{4130024167189}a^{7}+\frac{11\!\cdots\!21}{4130024167189}a^{6}-\frac{52\!\cdots\!44}{4130024167189}a^{5}-\frac{53\!\cdots\!28}{4130024167189}a^{4}+\frac{19\!\cdots\!44}{4130024167189}a^{3}+\frac{955496576923883}{4130024167189}a^{2}-\frac{209733269379739}{4130024167189}a-\frac{30248746610620}{4130024167189}$, $\frac{2890511358063}{4130024167189}a^{17}-\frac{1560162632522}{4130024167189}a^{16}-\frac{73122423633995}{4130024167189}a^{15}+\frac{24388276739092}{4130024167189}a^{14}+\frac{685881029448391}{4130024167189}a^{13}-\frac{86672546094962}{4130024167189}a^{12}-\frac{30\!\cdots\!42}{4130024167189}a^{11}-\frac{162253497367034}{4130024167189}a^{10}+\frac{67\!\cdots\!57}{4130024167189}a^{9}+\frac{11\!\cdots\!57}{4130024167189}a^{8}-\frac{79\!\cdots\!98}{4130024167189}a^{7}-\frac{17\!\cdots\!78}{4130024167189}a^{6}+\frac{47\!\cdots\!47}{4130024167189}a^{5}+\frac{965705065169861}{4130024167189}a^{4}-\frac{12\!\cdots\!20}{4130024167189}a^{3}-\frac{138159267703595}{4130024167189}a^{2}+\frac{109442563121606}{4130024167189}a-\frac{1237844832770}{4130024167189}$, $\frac{451949014783}{4130024167189}a^{17}+\frac{1639263095048}{4130024167189}a^{16}-\frac{14275208809197}{4130024167189}a^{15}-\frac{41563471765725}{4130024167189}a^{14}+\frac{167533752824547}{4130024167189}a^{13}+\frac{386175519383111}{4130024167189}a^{12}-\frac{924193241138791}{4130024167189}a^{11}-\frac{16\!\cdots\!89}{4130024167189}a^{10}+\frac{25\!\cdots\!34}{4130024167189}a^{9}+\frac{34\!\cdots\!43}{4130024167189}a^{8}-\frac{36\!\cdots\!04}{4130024167189}a^{7}-\frac{35\!\cdots\!64}{4130024167189}a^{6}+\frac{26\!\cdots\!64}{4130024167189}a^{5}+\frac{16\!\cdots\!90}{4130024167189}a^{4}-\frac{862580466636812}{4130024167189}a^{3}-\frac{237259060788037}{4130024167189}a^{2}+\frac{83802671713975}{4130024167189}a+\frac{151011838219}{4130024167189}$, $\frac{6685840668783}{4130024167189}a^{17}-\frac{5641323340899}{4130024167189}a^{16}-\frac{167341205573587}{4130024167189}a^{15}+\frac{105866765434810}{4130024167189}a^{14}+\frac{15\!\cdots\!16}{4130024167189}a^{13}-\frac{636975400699115}{4130024167189}a^{12}-\frac{67\!\cdots\!61}{4130024167189}a^{11}+\frac{13\!\cdots\!73}{4130024167189}a^{10}+\frac{15\!\cdots\!43}{4130024167189}a^{9}-\frac{558214499242677}{4130024167189}a^{8}-\frac{17\!\cdots\!29}{4130024167189}a^{7}-\frac{14\!\cdots\!34}{4130024167189}a^{6}+\frac{10\!\cdots\!88}{4130024167189}a^{5}+\frac{15\!\cdots\!99}{4130024167189}a^{4}-\frac{28\!\cdots\!48}{4130024167189}a^{3}-\frac{455003890791897}{4130024167189}a^{2}+\frac{225556607112622}{4130024167189}a+\frac{22540791159650}{4130024167189}$, $\frac{2677134831686}{4130024167189}a^{17}-\frac{5601280229469}{4130024167189}a^{16}-\frac{62578418722695}{4130024167189}a^{15}+\frac{123433177141650}{4130024167189}a^{14}+\frac{529917242239759}{4130024167189}a^{13}-\frac{973967141693871}{4130024167189}a^{12}-\frac{20\!\cdots\!72}{4130024167189}a^{11}+\frac{34\!\cdots\!33}{4130024167189}a^{10}+\frac{39\!\cdots\!17}{4130024167189}a^{9}-\frac{61\!\cdots\!13}{4130024167189}a^{8}-\frac{40\!\cdots\!58}{4130024167189}a^{7}+\frac{53\!\cdots\!34}{4130024167189}a^{6}+\frac{20\!\cdots\!85}{4130024167189}a^{5}-\frac{20\!\cdots\!00}{4130024167189}a^{4}-\frac{518334616484546}{4130024167189}a^{3}+\frac{219410090710498}{4130024167189}a^{2}+\frac{29266878560742}{4130024167189}a-\frac{2519766968019}{4130024167189}$
| 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: | \( 26179751.3516 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 26179751.3516 \cdot 1}{2\cdot\sqrt{563362135874260093126953125}}\cr\approx \mathstrut & 0.144571143647 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 25*x^16 + 20*x^15 + 232*x^14 - 137*x^13 - 1018*x^12 + 403*x^11 + 2291*x^10 - 566*x^9 - 2742*x^8 + 387*x^7 + 1711*x^6 - 131*x^5 - 505*x^4 + 30*x^3 + 55*x^2 - 5*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 - 25*x^16 + 20*x^15 + 232*x^14 - 137*x^13 - 1018*x^12 + 403*x^11 + 2291*x^10 - 566*x^9 - 2742*x^8 + 387*x^7 + 1711*x^6 - 131*x^5 - 505*x^4 + 30*x^3 + 55*x^2 - 5*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 - 25*x^16 + 20*x^15 + 232*x^14 - 137*x^13 - 1018*x^12 + 403*x^11 + 2291*x^10 - 566*x^9 - 2742*x^8 + 387*x^7 + 1711*x^6 - 131*x^5 - 505*x^4 + 30*x^3 + 55*x^2 - 5*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 - 25*x^16 + 20*x^15 + 232*x^14 - 137*x^13 - 1018*x^12 + 403*x^11 + 2291*x^10 - 566*x^9 - 2742*x^8 + 387*x^7 + 1711*x^6 - 131*x^5 - 505*x^4 + 30*x^3 + 55*x^2 - 5*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
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
| \(\Q(\sqrt{5}) \), 3.3.361.1, 6.6.16290125.1, \(\Q(\zeta_{19})^+\) |
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 | $18$ | $18$ | R | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | $18$ | $18$ | R | $18$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | $18$ | $18$ | $18$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.18.9.1 | $x^{18} + 180 x^{17} + 14445 x^{16} + 679200 x^{15} + 20664900 x^{14} + 423486000 x^{13} + 5887570504 x^{12} + 54397260480 x^{11} + 316143109712 x^{10} + 1034969211206 x^{9} + 1580754753720 x^{8} + 1360718216520 x^{7} + 746510415004 x^{6} + 357128191140 x^{5} + 552895560364 x^{4} + 1314509471572 x^{3} + 1121303668936 x^{2} + 1315877100296 x + 1500010785049$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |
|
\(19\)
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |