Properties

Label 20.4.493...664.2
Degree $20$
Signature $[4, 8]$
Discriminant $4.934\times 10^{25}$
Root discriminant \(19.26\)
Ramified primes $2,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^4:C_5$ (as 20T23)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 14*x^17 - 36*x^16 - 24*x^15 + 76*x^14 - 174*x^13 - 197*x^12 + 210*x^11 + 34*x^10 - 176*x^9 + 19*x^8 - 58*x^7 - 100*x^6 + 16*x^5 + 13*x^4 - 2*x^3 + 10*x^2 + 4*x - 1)
 
gp: K = bnfinit(y^20 - 2*y^19 + 14*y^17 - 36*y^16 - 24*y^15 + 76*y^14 - 174*y^13 - 197*y^12 + 210*y^11 + 34*y^10 - 176*y^9 + 19*y^8 - 58*y^7 - 100*y^6 + 16*y^5 + 13*y^4 - 2*y^3 + 10*y^2 + 4*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 14*x^17 - 36*x^16 - 24*x^15 + 76*x^14 - 174*x^13 - 197*x^12 + 210*x^11 + 34*x^10 - 176*x^9 + 19*x^8 - 58*x^7 - 100*x^6 + 16*x^5 + 13*x^4 - 2*x^3 + 10*x^2 + 4*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 14*x^17 - 36*x^16 - 24*x^15 + 76*x^14 - 174*x^13 - 197*x^12 + 210*x^11 + 34*x^10 - 176*x^9 + 19*x^8 - 58*x^7 - 100*x^6 + 16*x^5 + 13*x^4 - 2*x^3 + 10*x^2 + 4*x - 1)
 

\( x^{20} - 2 x^{19} + 14 x^{17} - 36 x^{16} - 24 x^{15} + 76 x^{14} - 174 x^{13} - 197 x^{12} + 210 x^{11} + 34 x^{10} - 176 x^{9} + 19 x^{8} - 58 x^{7} - 100 x^{6} + 16 x^{5} + 13 x^{4} + \cdots - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(49338146756019243307761664\) \(\medspace = 2^{30}\cdot 11^{16}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(19.26\)
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:  $2^{15/8}11^{4/5}\approx 24.977294240287762$
Ramified primes:   \(2\), \(11\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11}a^{13}+\frac{5}{11}a^{12}-\frac{2}{11}a^{11}+\frac{1}{11}a^{10}-\frac{1}{11}a^{8}+\frac{1}{11}a^{7}+\frac{3}{11}a^{5}-\frac{2}{11}a^{4}+\frac{2}{11}a^{3}-\frac{2}{11}a^{2}-\frac{2}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{14}-\frac{5}{11}a^{12}-\frac{5}{11}a^{10}-\frac{1}{11}a^{9}-\frac{5}{11}a^{8}-\frac{5}{11}a^{7}+\frac{3}{11}a^{6}+\frac{5}{11}a^{5}+\frac{1}{11}a^{4}-\frac{1}{11}a^{3}-\frac{3}{11}a^{2}+\frac{1}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{15}+\frac{3}{11}a^{12}-\frac{4}{11}a^{11}+\frac{4}{11}a^{10}-\frac{5}{11}a^{9}+\frac{1}{11}a^{8}-\frac{3}{11}a^{7}+\frac{5}{11}a^{6}+\frac{5}{11}a^{5}-\frac{4}{11}a^{3}+\frac{2}{11}a^{2}+\frac{2}{11}a-\frac{1}{11}$, $\frac{1}{11}a^{16}+\frac{3}{11}a^{12}-\frac{1}{11}a^{11}+\frac{3}{11}a^{10}+\frac{1}{11}a^{9}+\frac{2}{11}a^{7}+\frac{5}{11}a^{6}+\frac{2}{11}a^{5}+\frac{2}{11}a^{4}-\frac{4}{11}a^{3}-\frac{3}{11}a^{2}+\frac{5}{11}a+\frac{5}{11}$, $\frac{1}{11}a^{17}-\frac{5}{11}a^{12}-\frac{2}{11}a^{11}-\frac{2}{11}a^{10}+\frac{5}{11}a^{8}+\frac{2}{11}a^{7}+\frac{2}{11}a^{6}+\frac{4}{11}a^{5}+\frac{2}{11}a^{4}+\frac{2}{11}a^{3}+\frac{5}{11}$, $\frac{1}{121}a^{18}+\frac{2}{121}a^{17}+\frac{4}{121}a^{16}+\frac{3}{121}a^{14}-\frac{1}{121}a^{13}-\frac{6}{121}a^{12}-\frac{29}{121}a^{11}-\frac{36}{121}a^{10}+\frac{39}{121}a^{9}-\frac{7}{121}a^{8}-\frac{52}{121}a^{7}+\frac{26}{121}a^{6}+\frac{56}{121}a^{5}-\frac{13}{121}a^{4}-\frac{18}{121}a^{3}+\frac{26}{121}a^{2}+\frac{20}{121}a-\frac{47}{121}$, $\frac{1}{33972148677841}a^{19}+\frac{117763882196}{33972148677841}a^{18}+\frac{413636325220}{33972148677841}a^{17}+\frac{1222272052372}{33972148677841}a^{16}-\frac{1145141287155}{33972148677841}a^{15}+\frac{401611167424}{33972148677841}a^{14}+\frac{1512310380232}{33972148677841}a^{13}-\frac{2289135694117}{33972148677841}a^{12}+\frac{9116397299993}{33972148677841}a^{11}+\frac{550831379436}{3088377152531}a^{10}+\frac{745377593097}{33972148677841}a^{9}+\frac{633333929291}{3088377152531}a^{8}-\frac{3156794737262}{33972148677841}a^{7}+\frac{13164755328393}{33972148677841}a^{6}+\frac{56376942606}{3088377152531}a^{5}-\frac{958196935655}{3088377152531}a^{4}+\frac{2860033257896}{33972148677841}a^{3}+\frac{12325588528600}{33972148677841}a^{2}+\frac{16534732396230}{33972148677841}a-\frac{16485588049519}{33972148677841}$ Copy content Toggle raw display

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

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{7887087777342}{33972148677841}a^{19}-\frac{23742169075395}{33972148677841}a^{18}+\frac{17502036723136}{33972148677841}a^{17}+\frac{105687996029426}{33972148677841}a^{16}-\frac{390798412906527}{33972148677841}a^{15}+\frac{115809511800072}{33972148677841}a^{14}+\frac{713193674551398}{33972148677841}a^{13}-\frac{19\!\cdots\!79}{33972148677841}a^{12}-\frac{72611012187047}{33972148677841}a^{11}+\frac{28\!\cdots\!77}{33972148677841}a^{10}-\frac{13\!\cdots\!37}{33972148677841}a^{9}-\frac{13\!\cdots\!70}{33972148677841}a^{8}+\frac{99328941754875}{3088377152531}a^{7}-\frac{499401326713761}{33972148677841}a^{6}-\frac{34686834542619}{33972148677841}a^{5}+\frac{532436345100947}{33972148677841}a^{4}-\frac{31260845957416}{33972148677841}a^{3}+\frac{953869028621}{3088377152531}a^{2}+\frac{20318190662028}{33972148677841}a-\frac{32896131180292}{33972148677841}$, $\frac{5023559847238}{33972148677841}a^{19}-\frac{12814763501714}{33972148677841}a^{18}+\frac{10683038619026}{33972148677841}a^{17}+\frac{56560226256602}{33972148677841}a^{16}-\frac{211012779029113}{33972148677841}a^{15}+\frac{46207899271851}{33972148677841}a^{14}+\frac{218992679502350}{33972148677841}a^{13}-\frac{96689973512585}{3088377152531}a^{12}-\frac{113656304506836}{33972148677841}a^{11}+\frac{479361621166475}{33972148677841}a^{10}-\frac{715816565416133}{33972148677841}a^{9}+\frac{375052624422956}{33972148677841}a^{8}+\frac{95941712441398}{33972148677841}a^{7}-\frac{89734391006615}{3088377152531}a^{6}-\frac{35790811842050}{33972148677841}a^{5}-\frac{32657680500839}{33972148677841}a^{4}-\frac{43161925200169}{33972148677841}a^{3}+\frac{143309667130031}{33972148677841}a^{2}+\frac{31538746612081}{33972148677841}a+\frac{8583793255866}{33972148677841}$, $\frac{7240323236098}{33972148677841}a^{19}-\frac{19222403320585}{33972148677841}a^{18}+\frac{8104479961979}{33972148677841}a^{17}+\frac{104645695157606}{33972148677841}a^{16}-\frac{329569294049359}{33972148677841}a^{15}-\frac{17406588853833}{33972148677841}a^{14}+\frac{715656919094928}{33972148677841}a^{13}-\frac{16\!\cdots\!00}{33972148677841}a^{12}-\frac{639023217014436}{33972148677841}a^{11}+\frac{26\!\cdots\!48}{33972148677841}a^{10}-\frac{671379157565289}{33972148677841}a^{9}-\frac{14\!\cdots\!57}{33972148677841}a^{8}+\frac{883638705933455}{33972148677841}a^{7}-\frac{661212219371699}{33972148677841}a^{6}-\frac{79796779275244}{33972148677841}a^{5}+\frac{711607622346914}{33972148677841}a^{4}-\frac{113399380189883}{33972148677841}a^{3}+\frac{81411457522852}{33972148677841}a^{2}+\frac{43907277463986}{33972148677841}a-\frac{51256834104876}{33972148677841}$, $\frac{1143870687911}{33972148677841}a^{19}+\frac{5499019366268}{33972148677841}a^{18}-\frac{20191893381013}{33972148677841}a^{17}+\frac{27530321188659}{33972148677841}a^{16}+\frac{62315475137659}{33972148677841}a^{15}-\frac{369159885335088}{33972148677841}a^{14}+\frac{94401991786442}{33972148677841}a^{13}+\frac{411022900174791}{33972148677841}a^{12}-\frac{19\!\cdots\!13}{33972148677841}a^{11}-\frac{355584704571483}{33972148677841}a^{10}+\frac{21\!\cdots\!96}{33972148677841}a^{9}-\frac{10\!\cdots\!22}{33972148677841}a^{8}-\frac{11\!\cdots\!98}{33972148677841}a^{7}+\frac{733260503788140}{33972148677841}a^{6}-\frac{753896611331625}{33972148677841}a^{5}-\frac{402443067864337}{33972148677841}a^{4}+\frac{294728813424413}{33972148677841}a^{3}-\frac{79601434932171}{33972148677841}a^{2}-\frac{41752107297101}{33972148677841}a+\frac{56944100687612}{33972148677841}$, $\frac{13708459747614}{33972148677841}a^{19}-\frac{28071475717649}{33972148677841}a^{18}+\frac{1708732302495}{33972148677841}a^{17}+\frac{16933763327948}{3088377152531}a^{16}-\frac{491388203447401}{33972148677841}a^{15}-\frac{303042267971239}{33972148677841}a^{14}+\frac{976368487801512}{33972148677841}a^{13}-\frac{22\!\cdots\!17}{33972148677841}a^{12}-\frac{24\!\cdots\!94}{33972148677841}a^{11}+\frac{25\!\cdots\!77}{33972148677841}a^{10}+\frac{11\!\cdots\!88}{33972148677841}a^{9}-\frac{16\!\cdots\!39}{33972148677841}a^{8}-\frac{840829565483151}{33972148677841}a^{7}-\frac{950692264338658}{33972148677841}a^{6}-\frac{443288211770612}{33972148677841}a^{5}+\frac{260596856249530}{33972148677841}a^{4}+\frac{295449525632963}{33972148677841}a^{3}+\frac{174843227129704}{33972148677841}a^{2}+\frac{15484229535595}{33972148677841}a-\frac{2380426760873}{3088377152531}$, $\frac{3930210462336}{33972148677841}a^{19}-\frac{11173744239612}{33972148677841}a^{18}+\frac{11269523314625}{33972148677841}a^{17}+\frac{44947583903407}{33972148677841}a^{16}-\frac{184880468815492}{33972148677841}a^{15}+\frac{7727483134892}{3088377152531}a^{14}+\frac{205081603905633}{33972148677841}a^{13}-\frac{10\!\cdots\!17}{33972148677841}a^{12}+\frac{95375992377058}{33972148677841}a^{11}+\frac{661882113093056}{33972148677841}a^{10}-\frac{12\!\cdots\!54}{33972148677841}a^{9}-\frac{70178316466776}{33972148677841}a^{8}+\frac{777086755469088}{33972148677841}a^{7}-\frac{599426400367342}{33972148677841}a^{6}-\frac{379730328046413}{33972148677841}a^{5}-\frac{120157896018187}{33972148677841}a^{4}-\frac{163017567373647}{33972148677841}a^{3}+\frac{6389163984633}{33972148677841}a^{2}+\frac{11243562986352}{3088377152531}a+\frac{59499480849233}{33972148677841}$, $\frac{674004260629}{33972148677841}a^{19}-\frac{3077693091764}{33972148677841}a^{18}+\frac{2914200976358}{33972148677841}a^{17}+\frac{11561222481887}{33972148677841}a^{16}-\frac{51109057416440}{33972148677841}a^{15}+\frac{40657602639084}{33972148677841}a^{14}+\frac{123447666145924}{33972148677841}a^{13}-\frac{276869602108386}{33972148677841}a^{12}+\frac{132760755193794}{33972148677841}a^{11}+\frac{5108890886919}{280761559321}a^{10}-\frac{423124710111222}{33972148677841}a^{9}-\frac{32056145925594}{3088377152531}a^{8}+\frac{389248605713642}{33972148677841}a^{7}+\frac{2793227257378}{33972148677841}a^{6}-\frac{198048210128}{280761559321}a^{5}+\frac{19301679687769}{3088377152531}a^{4}-\frac{74027986206409}{33972148677841}a^{3}-\frac{100667828648917}{33972148677841}a^{2}+\frac{1321483839209}{33972148677841}a-\frac{26896467726445}{33972148677841}$, $\frac{12715194846701}{33972148677841}a^{19}-\frac{32192342315060}{33972148677841}a^{18}+\frac{14157733937779}{33972148677841}a^{17}+\frac{174482462084031}{33972148677841}a^{16}-\frac{545799903356382}{33972148677841}a^{15}-\frac{56333063446428}{33972148677841}a^{14}+\frac{10\!\cdots\!75}{33972148677841}a^{13}-\frac{26\!\cdots\!50}{33972148677841}a^{12}-\frac{12\!\cdots\!50}{33972148677841}a^{11}+\frac{36\!\cdots\!37}{33972148677841}a^{10}-\frac{557727102303080}{33972148677841}a^{9}-\frac{21\!\cdots\!76}{33972148677841}a^{8}+\frac{417428219564090}{33972148677841}a^{7}-\frac{669207381018069}{33972148677841}a^{6}-\frac{185818185040318}{33972148677841}a^{5}+\frac{403234924270542}{33972148677841}a^{4}-\frac{10801365727555}{33972148677841}a^{3}+\frac{10602102019103}{33972148677841}a^{2}-\frac{65912974617014}{33972148677841}a-\frac{23296280707608}{33972148677841}$, $\frac{8220160487398}{33972148677841}a^{19}-\frac{20576164041306}{33972148677841}a^{18}+\frac{17030784202246}{33972148677841}a^{17}+\frac{93361307515616}{33972148677841}a^{16}-\frac{340577921365874}{33972148677841}a^{15}+\frac{61210681420079}{33972148677841}a^{14}+\frac{359193108876995}{33972148677841}a^{13}-\frac{17\!\cdots\!40}{33972148677841}a^{12}-\frac{349196216834892}{33972148677841}a^{11}+\frac{734518254487518}{33972148677841}a^{10}-\frac{12\!\cdots\!71}{33972148677841}a^{9}+\frac{569387845634}{33972148677841}a^{8}+\frac{97476210857479}{33972148677841}a^{7}-\frac{11\!\cdots\!58}{33972148677841}a^{6}-\frac{338115558591626}{33972148677841}a^{5}-\frac{469713344828328}{33972148677841}a^{4}-\frac{212840465341180}{33972148677841}a^{3}-\frac{24730121411354}{33972148677841}a^{2}-\frac{13412986362258}{33972148677841}a+\frac{9838911706090}{33972148677841}$, $\frac{2885834382345}{33972148677841}a^{19}-\frac{1799134874271}{33972148677841}a^{18}-\frac{12167558197340}{33972148677841}a^{17}+\frac{53333203456118}{33972148677841}a^{16}-\frac{59450364085326}{33972148677841}a^{15}-\frac{265567973765542}{33972148677841}a^{14}+\frac{334960303227771}{33972148677841}a^{13}-\frac{288664864890261}{33972148677841}a^{12}-\frac{15\!\cdots\!00}{33972148677841}a^{11}+\frac{886690435743405}{33972148677841}a^{10}+\frac{826615940332794}{33972148677841}a^{9}-\frac{16\!\cdots\!12}{33972148677841}a^{8}+\frac{252417980030999}{33972148677841}a^{7}+\frac{286075962494679}{33972148677841}a^{6}-\frac{10\!\cdots\!13}{33972148677841}a^{5}+\frac{107116654351323}{33972148677841}a^{4}+\frac{4667927554256}{3088377152531}a^{3}-\frac{137926595309680}{33972148677841}a^{2}+\frac{81365383790832}{33972148677841}a+\frac{36030337921958}{33972148677841}$, $\frac{11053159271431}{33972148677841}a^{19}-\frac{23194992208676}{33972148677841}a^{18}+\frac{776762352813}{33972148677841}a^{17}+\frac{157688633364799}{33972148677841}a^{16}-\frac{413620068987647}{33972148677841}a^{15}-\frac{244264259443873}{33972148677841}a^{14}+\frac{915007596820619}{33972148677841}a^{13}-\frac{179658469345770}{3088377152531}a^{12}-\frac{20\!\cdots\!61}{33972148677841}a^{11}+\frac{27\!\cdots\!28}{33972148677841}a^{10}+\frac{421824643650899}{33972148677841}a^{9}-\frac{22\!\cdots\!48}{33972148677841}a^{8}+\frac{142573475533699}{33972148677841}a^{7}-\frac{31983513899166}{3088377152531}a^{6}-\frac{814345337600310}{33972148677841}a^{5}+\frac{169302247665560}{33972148677841}a^{4}+\frac{191853711723504}{33972148677841}a^{3}+\frac{14646167684626}{33972148677841}a^{2}+\frac{86685318107295}{33972148677841}a+\frac{49775173052788}{33972148677841}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 90919.8145635 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 90919.8145635 \cdot 1}{2\cdot\sqrt{49338146756019243307761664}}\cr\approx \mathstrut & 0.251533630318 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 14*x^17 - 36*x^16 - 24*x^15 + 76*x^14 - 174*x^13 - 197*x^12 + 210*x^11 + 34*x^10 - 176*x^9 + 19*x^8 - 58*x^7 - 100*x^6 + 16*x^5 + 13*x^4 - 2*x^3 + 10*x^2 + 4*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^20 - 2*x^19 + 14*x^17 - 36*x^16 - 24*x^15 + 76*x^14 - 174*x^13 - 197*x^12 + 210*x^11 + 34*x^10 - 176*x^9 + 19*x^8 - 58*x^7 - 100*x^6 + 16*x^5 + 13*x^4 - 2*x^3 + 10*x^2 + 4*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^20 - 2*x^19 + 14*x^17 - 36*x^16 - 24*x^15 + 76*x^14 - 174*x^13 - 197*x^12 + 210*x^11 + 34*x^10 - 176*x^9 + 19*x^8 - 58*x^7 - 100*x^6 + 16*x^5 + 13*x^4 - 2*x^3 + 10*x^2 + 4*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^20 - 2*x^19 + 14*x^17 - 36*x^16 - 24*x^15 + 76*x^14 - 174*x^13 - 197*x^12 + 210*x^11 + 34*x^10 - 176*x^9 + 19*x^8 - 58*x^7 - 100*x^6 + 16*x^5 + 13*x^4 - 2*x^3 + 10*x^2 + 4*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

$C_2^4:C_5$ (as 20T23):

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 80
The 8 conjugacy class representatives for $C_2^4:C_5$
Character table for $C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.219503494144.1, 10.2.219503494144.1, 10.6.219503494144.2

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 10 siblings: data not computed
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 10.2.219503494144.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.5.0.1}{5} }^{4}$ ${\href{/padicField/5.5.0.1}{5} }^{4}$ ${\href{/padicField/7.5.0.1}{5} }^{4}$ R ${\href{/padicField/13.5.0.1}{5} }^{4}$ ${\href{/padicField/17.5.0.1}{5} }^{4}$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.5.0.1}{5} }^{4}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $20$$4$$5$$30$
\(11\) Copy content Toggle raw display 11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$