↑

Properties

Label 20.0.19590707183...3125.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{31}\cdot 29^{10}$
Root discriminant $65.25$
Ramified primes $5, 29$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T9)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35445196901, 53878854080, 43037450335, 21092371985, 5808953160, -270425399, -1008503770, -538651975, -138381605, -16162825, 6716306, 3023350, 934495, 92155, 17495, -2329, 840, -80, 40, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 40*x^18 - 80*x^17 + 840*x^16 - 2329*x^15 + 17495*x^14 + 92155*x^13 + 934495*x^12 + 3023350*x^11 + 6716306*x^10 - 16162825*x^9 - 138381605*x^8 - 538651975*x^7 - 1008503770*x^6 - 270425399*x^5 + 5808953160*x^4 + 21092371985*x^3 + 43037450335*x^2 + 53878854080*x + 35445196901)
 
gp: K = bnfinit(x^20 - 5*x^19 + 40*x^18 - 80*x^17 + 840*x^16 - 2329*x^15 + 17495*x^14 + 92155*x^13 + 934495*x^12 + 3023350*x^11 + 6716306*x^10 - 16162825*x^9 - 138381605*x^8 - 538651975*x^7 - 1008503770*x^6 - 270425399*x^5 + 5808953160*x^4 + 21092371985*x^3 + 43037450335*x^2 + 53878854080*x + 35445196901, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 40 x^{18} - 80 x^{17} + 840 x^{16} - 2329 x^{15} + 17495 x^{14} + 92155 x^{13} + 934495 x^{12} + 3023350 x^{11} + 6716306 x^{10} - 16162825 x^{9} - 138381605 x^{8} - 538651975 x^{7} - 1008503770 x^{6} - 270425399 x^{5} + 5808953160 x^{4} + 21092371985 x^{3} + 43037450335 x^{2} + 53878854080 x + 35445196901 \)

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:  \(1959070718382489867508411407470703125=5^{31}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.25$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{112} a^{18} + \frac{1}{28} a^{17} + \frac{1}{112} a^{16} + \frac{3}{16} a^{15} + \frac{1}{56} a^{14} - \frac{11}{56} a^{13} + \frac{11}{112} a^{12} - \frac{1}{14} a^{11} + \frac{11}{56} a^{10} - \frac{5}{28} a^{9} - \frac{9}{28} a^{8} - \frac{7}{16} a^{7} - \frac{3}{8} a^{6} + \frac{17}{56} a^{5} + \frac{23}{56} a^{4} + \frac{7}{16} a^{3} + \frac{39}{112} a^{2} - \frac{5}{16} a + \frac{55}{112}$, $\frac{1}{148952928389448747655886697275181613907889581803058268112919971918838603299779851374275002992} a^{19} - \frac{91533955452414811398681306050529474647887924521486551782087905247874849850532117092380403}{148952928389448747655886697275181613907889581803058268112919971918838603299779851374275002992} a^{18} + \frac{11871071130094629439439496945262299815344639559457720003592043700677920856724335049547451045}{148952928389448747655886697275181613907889581803058268112919971918838603299779851374275002992} a^{17} - \frac{231905070982059832215291501174615900540119453793464525572358908925869233922121507998183965}{10639494884960624832563335519655829564849255843075590579494283708488471664269989383876785928} a^{16} + \frac{37073155790822544355841294335201024927795421417119795780386319577539853945410468358260260867}{148952928389448747655886697275181613907889581803058268112919971918838603299779851374275002992} a^{15} + \frac{3487336949598642058495533285021783397236094989255821276435767511538542006716586050014126903}{18619116048681093456985837159397701738486197725382283514114996489854825412472481421784375374} a^{14} + \frac{22641788571475231934279247810135137767636166292732521687109415642386718588581433152323902233}{148952928389448747655886697275181613907889581803058268112919971918838603299779851374275002992} a^{13} - \frac{33304069765064731705185556393033730618604530964460398876466779083238730587086888881872142197}{148952928389448747655886697275181613907889581803058268112919971918838603299779851374275002992} a^{12} - \frac{1791544550579274193681341512967365071936056685855158425055392549038900771365071309092891759}{74476464194724373827943348637590806953944790901529134056459985959419301649889925687137501496} a^{11} - \frac{16438928585078970827495952083717154517809539494309487147027844424384428596990829031243291799}{74476464194724373827943348637590806953944790901529134056459985959419301649889925687137501496} a^{10} + \frac{12170463186436745199825864721485616939094447916233930979816200319356284019082898764682995605}{37238232097362186913971674318795403476972395450764567028229992979709650824944962843568750748} a^{9} + \frac{4753630017307374930230252629040318519292197673489093212953776717071362789076096706889723857}{21278989769921249665126671039311659129698511686151181158988567416976943328539978767753571856} a^{8} + \frac{1664743206562524940175261992680447349691143189428260031143465928526893501609951211269562859}{21278989769921249665126671039311659129698511686151181158988567416976943328539978767753571856} a^{7} + \frac{10268050226830578429713997685055149640975328450891691216588610180447772858860067146927680051}{37238232097362186913971674318795403476972395450764567028229992979709650824944962843568750748} a^{6} - \frac{2535348948932264128369815525890939201817830753249441616261173918909450452139150700075287034}{9309558024340546728492918579698850869243098862691141757057498244927412706236240710892187687} a^{5} + \frac{6601822632621553636231130319340883283980595458215731648333849632792803877702594835262239045}{21278989769921249665126671039311659129698511686151181158988567416976943328539978767753571856} a^{4} + \frac{797767921724583689064790689959869920567789759550585297027923118973569488311377458439853677}{37238232097362186913971674318795403476972395450764567028229992979709650824944962843568750748} a^{3} + \frac{588341484900166851522327752527612071708581681367750633953705344555496742518936816228576363}{5319747442480312416281667759827914782424627921537795289747141854244235832134994691938392964} a^{2} + \frac{1236100586533335313427185955196785645328428369416951033283250676073739223329251697415849659}{37238232097362186913971674318795403476972395450764567028229992979709650824944962843568750748} a - \frac{6581988521605991597918629344768764225154395911055268797611658325159066744241667318286768787}{21278989769921249665126671039311659129698511686151181158988567416976943328539978767753571856}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.105125.2, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$