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Properties

Label 20.0.45726371671...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{17}\cdot 5^{37}\cdot 11^{10}\cdot 31^{5}$
Root discriminant $680.78$
Ramified primes $2, 3, 5, 11, 31$
Class number $80$ (GRH)
Class group $[2, 2, 20]$ (GRH)
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![197430725614375, 41896992259375, -26823631243750, -248531078125, 856847774500, -439041788393, 168604014460, -14123682335, -1244384545, -259765920, 26044322, 11523240, 5614535, -37205, 58720, -14471, -110, 35, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 20*x^18 + 35*x^17 - 110*x^16 - 14471*x^15 + 58720*x^14 - 37205*x^13 + 5614535*x^12 + 11523240*x^11 + 26044322*x^10 - 259765920*x^9 - 1244384545*x^8 - 14123682335*x^7 + 168604014460*x^6 - 439041788393*x^5 + 856847774500*x^4 - 248531078125*x^3 - 26823631243750*x^2 + 41896992259375*x + 197430725614375)
 
gp: K = bnfinit(x^20 - 5*x^19 + 20*x^18 + 35*x^17 - 110*x^16 - 14471*x^15 + 58720*x^14 - 37205*x^13 + 5614535*x^12 + 11523240*x^11 + 26044322*x^10 - 259765920*x^9 - 1244384545*x^8 - 14123682335*x^7 + 168604014460*x^6 - 439041788393*x^5 + 856847774500*x^4 - 248531078125*x^3 - 26823631243750*x^2 + 41896992259375*x + 197430725614375, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 20 x^{18} + 35 x^{17} - 110 x^{16} - 14471 x^{15} + 58720 x^{14} - 37205 x^{13} + 5614535 x^{12} + 11523240 x^{11} + 26044322 x^{10} - 259765920 x^{9} - 1244384545 x^{8} - 14123682335 x^{7} + 168604014460 x^{6} - 439041788393 x^{5} + 856847774500 x^{4} - 248531078125 x^{3} - 26823631243750 x^{2} + 41896992259375 x + 197430725614375 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(457263716717479671795708718299865722656250000000000000000=2^{16}\cdot 3^{17}\cdot 5^{37}\cdot 11^{10}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $680.78$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 11, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{25} a^{7} + \frac{2}{25} a^{6} - \frac{1}{25} a^{5} - \frac{2}{25} a^{4} + \frac{1}{25} a^{3}$, $\frac{1}{25} a^{8} + \frac{8}{25} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{9} - \frac{2}{25} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{125} a^{10} - \frac{2}{125} a^{9} + \frac{2}{25} a^{6} - \frac{2}{125} a^{5} - \frac{1}{125} a^{4} - \frac{9}{25} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{125} a^{11} + \frac{1}{125} a^{9} - \frac{2}{125} a^{6} + \frac{1}{25} a^{5} - \frac{12}{125} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{625} a^{12} + \frac{1}{625} a^{11} - \frac{1}{625} a^{10} - \frac{1}{125} a^{9} + \frac{2}{125} a^{8} + \frac{3}{625} a^{7} + \frac{43}{625} a^{6} - \frac{33}{625} a^{5} + \frac{2}{25} a^{4} - \frac{1}{5} a^{2}$, $\frac{1}{625} a^{13} - \frac{2}{625} a^{11} + \frac{1}{625} a^{10} + \frac{1}{125} a^{9} - \frac{7}{625} a^{8} - \frac{2}{125} a^{7} - \frac{1}{625} a^{6} - \frac{2}{625} a^{5} + \frac{9}{125} a^{4} + \frac{9}{25} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{3125} a^{14} + \frac{2}{3125} a^{13} - \frac{1}{3125} a^{12} - \frac{7}{3125} a^{11} + \frac{6}{3125} a^{10} + \frac{18}{3125} a^{9} + \frac{36}{3125} a^{8} + \frac{7}{3125} a^{7} - \frac{151}{3125} a^{6} - \frac{167}{3125} a^{5} - \frac{2}{125} a^{4} + \frac{11}{25} a^{3} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{18750} a^{15} + \frac{1}{9375} a^{14} - \frac{1}{3125} a^{13} - \frac{7}{18750} a^{12} - \frac{17}{9375} a^{11} + \frac{19}{9375} a^{10} + \frac{6}{3125} a^{9} - \frac{104}{9375} a^{8} - \frac{113}{9375} a^{7} + \frac{198}{3125} a^{6} + \frac{41}{1875} a^{5} + \frac{22}{375} a^{4} - \frac{1}{30} a^{3} + \frac{1}{5} a^{2} + \frac{1}{15} a - \frac{1}{6}$, $\frac{1}{2062500} a^{16} + \frac{53}{2062500} a^{15} - \frac{3}{68750} a^{14} - \frac{23}{412500} a^{13} + \frac{43}{412500} a^{12} + \frac{1993}{1031250} a^{11} + \frac{259}{171875} a^{10} - \frac{2107}{206250} a^{9} + \frac{1759}{103125} a^{8} + \frac{1239}{68750} a^{7} - \frac{4483}{93750} a^{6} + \frac{36811}{1031250} a^{5} + \frac{89}{1500} a^{4} - \frac{321}{1100} a^{3} - \frac{29}{165} a^{2} - \frac{79}{300} a - \frac{29}{100}$, $\frac{1}{10312500} a^{17} + \frac{1}{10312500} a^{16} - \frac{8}{859375} a^{15} + \frac{1}{62500} a^{14} - \frac{247}{687500} a^{13} + \frac{149}{234375} a^{12} - \frac{6766}{2578125} a^{11} - \frac{5881}{1718750} a^{10} - \frac{898}{171875} a^{9} - \frac{19939}{1031250} a^{8} - \frac{18661}{1718750} a^{7} + \frac{94829}{1718750} a^{6} - \frac{288269}{10312500} a^{5} + \frac{4513}{82500} a^{4} - \frac{237}{550} a^{3} + \frac{4871}{16500} a^{2} + \frac{107}{500} a - \frac{313}{750}$, $\frac{1}{10312500} a^{18} - \frac{1}{5156250} a^{16} - \frac{17}{859375} a^{15} - \frac{16}{515625} a^{14} - \frac{991}{2578125} a^{13} + \frac{147}{687500} a^{12} - \frac{1396}{859375} a^{11} - \frac{20617}{5156250} a^{10} - \frac{1906}{171875} a^{9} - \frac{17073}{859375} a^{8} - \frac{15821}{1031250} a^{7} - \frac{861313}{10312500} a^{6} - \frac{20518}{859375} a^{5} + \frac{977}{13750} a^{4} + \frac{653}{1375} a^{3} - \frac{17}{825} a^{2} + \frac{37}{375} a + \frac{187}{500}$, $\frac{1}{418023457526780815278328922495275194450867451303847232146737587636573861717047329687500} a^{19} - \frac{76084618033938212994110064123554712375807087787774571502776448092211610338531}{23223525418154489737684940138626399691714858405769290674818754868698547873169296093750} a^{18} - \frac{6359383220253092594456980227229063692504437490182803924415870201186622032053523}{209011728763390407639164461247637597225433725651923616073368793818286930858523664843750} a^{17} - \frac{1869285078550840945214373412477570420250303176337851953963556807331906753542417}{11611762709077244868842470069313199845857429202884645337409377434349273936584648046875} a^{16} - \frac{3537666204151647310946365105811266643884991406521004961641434182499668817701574377}{209011728763390407639164461247637597225433725651923616073368793818286930858523664843750} a^{15} + \frac{412529783558119112904525544507855026863778099674835243093660730372169322903778726}{11611762709077244868842470069313199845857429202884645337409377434349273936584648046875} a^{14} + \frac{247359146578294102298150334473912272421942350503521466723553962869399206790523069037}{418023457526780815278328922495275194450867451303847232146737587636573861717047329687500} a^{13} + \frac{10251559219938262331114441330284093806767841331047650401529529954615112084788282862}{34835288127231734606527410207939599537572287608653936012228132303047821809753944140625} a^{12} + \frac{779996983083591520620679875485557153318260867093500296127797636629892871102039359159}{209011728763390407639164461247637597225433725651923616073368793818286930858523664843750} a^{11} + \frac{30518273660237301448327511987911443572355796206102848798313315147930280393201400324}{9500533125608654892689293693074436237519714802360164366971308809922133220841984765625} a^{10} - \frac{1742791510678323471983550247590368319107822725409710950711374593852865089204307556344}{104505864381695203819582230623818798612716862825961808036684396909143465429261832421875} a^{9} - \frac{1510863534814759463691303139305164609604709287321812331962516931677489145572321395271}{209011728763390407639164461247637597225433725651923616073368793818286930858523664843750} a^{8} - \frac{81667197874402151355535798264260484938454660210767760939219270095146805395789116843}{139341152508926938426109640831758398150289150434615744048912529212191287239015776562500} a^{7} - \frac{6143399254530329193644715745296530082193836692070944012290003920696544973945046024319}{209011728763390407639164461247637597225433725651923616073368793818286930858523664843750} a^{6} + \frac{617137111836410353784743484605286572973361684465929179886673503667691284714038273703}{23223525418154489737684940138626399691714858405769290674818754868698547873169296093750} a^{5} - \frac{18469657318687146322023569122603074998148961454366450947992999177897738534545367652}{836046915053561630556657844990550388901734902607694464293475175273147723434094659375} a^{4} - \frac{1382517727754226499487491560507961613762625211847732064345106015725655510353280243}{10133902000649231885535246605946065320021029122517508658102729397250275435564783750} a^{3} + \frac{41686387699074661619791173095350409020648362782642450348466781827015483297688246106}{167209383010712326111331568998110077780346980521538892858695035054629544686818931875} a^{2} - \frac{7681603277271323343811087032421743226864930190712258585901230815074036559885882509}{20267804001298463771070493211892130640042058245035017316205458794500550871129567500} a - \frac{7291856904484754807781440721845620104660838870537545021442540778154202002893118646}{15200853000973847828302869908919097980031543683776262987154094095875413153347175625}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{20}$, which has order $80$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 16621017784297850000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.56265.4, 5.1.2531250000.1, 10.0.1031890245117187500000000.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.5.9.5$x^{5} + 105$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.5$x^{5} + 105$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.19$x^{10} + 110$$10$$1$$19$$F_{5}\times C_2$$[9/4]_{4}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
$31$31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$