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Properties

Label 20.0.25297701006...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{33}\cdot 3^{16}\cdot 5^{15}\cdot 7^{18}\cdot 11^{18}\cdot 19^{5}$
Root discriminant $2631.20$
Ramified primes $2, 3, 5, 7, 11, 19$
Class number Not computed
Class group Not computed
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![1231798969663440772496, 356450589632390596448, 33900336099376453616, -788955503763961632, -1007444850845660192, -143870078913265600, -1573676037978240, 1319017496324416, 148664630999080, 10739485365224, -177532729296, -108539004640, -2922822172, -794403800, 22330392, -1970920, 343364, 1812, 934, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 934*x^18 + 1812*x^17 + 343364*x^16 - 1970920*x^15 + 22330392*x^14 - 794403800*x^13 - 2922822172*x^12 - 108539004640*x^11 - 177532729296*x^10 + 10739485365224*x^9 + 148664630999080*x^8 + 1319017496324416*x^7 - 1573676037978240*x^6 - 143870078913265600*x^5 - 1007444850845660192*x^4 - 788955503763961632*x^3 + 33900336099376453616*x^2 + 356450589632390596448*x + 1231798969663440772496)
 
gp: K = bnfinit(x^20 - 10*x^19 + 934*x^18 + 1812*x^17 + 343364*x^16 - 1970920*x^15 + 22330392*x^14 - 794403800*x^13 - 2922822172*x^12 - 108539004640*x^11 - 177532729296*x^10 + 10739485365224*x^9 + 148664630999080*x^8 + 1319017496324416*x^7 - 1573676037978240*x^6 - 143870078913265600*x^5 - 1007444850845660192*x^4 - 788955503763961632*x^3 + 33900336099376453616*x^2 + 356450589632390596448*x + 1231798969663440772496, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 934 x^{18} + 1812 x^{17} + 343364 x^{16} - 1970920 x^{15} + 22330392 x^{14} - 794403800 x^{13} - 2922822172 x^{12} - 108539004640 x^{11} - 177532729296 x^{10} + 10739485365224 x^{9} + 148664630999080 x^{8} + 1319017496324416 x^{7} - 1573676037978240 x^{6} - 143870078913265600 x^{5} - 1007444850845660192 x^{4} - 788955503763961632 x^{3} + 33900336099376453616 x^{2} + 356450589632390596448 x + 1231798969663440772496 \)

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:  \(252977010060091794322686019356947527998852560912646144000000000000000=2^{33}\cdot 3^{16}\cdot 5^{15}\cdot 7^{18}\cdot 11^{18}\cdot 19^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2631.20$
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:  $2, 3, 5, 7, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{14} a^{8} + \frac{3}{14} a^{7} - \frac{1}{14} a^{6} + \frac{3}{14} a^{5} - \frac{3}{7} a^{4}$, $\frac{1}{28} a^{9} + \frac{1}{7} a^{7} + \frac{3}{14} a^{6} + \frac{3}{14} a^{5} - \frac{5}{14} a^{4}$, $\frac{1}{28} a^{10} - \frac{3}{14} a^{7} - \frac{1}{7} a^{6} + \frac{3}{14} a^{5} - \frac{1}{7} a^{4}$, $\frac{1}{28} a^{11} - \frac{2}{7} a^{4}$, $\frac{1}{140} a^{12} + \frac{1}{70} a^{11} - \frac{1}{70} a^{10} + \frac{1}{35} a^{8} + \frac{6}{35} a^{7} + \frac{1}{35} a^{6} - \frac{11}{70} a^{5} - \frac{1}{35} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{280} a^{13} + \frac{1}{70} a^{11} + \frac{1}{70} a^{10} + \frac{1}{70} a^{9} - \frac{1}{70} a^{8} + \frac{9}{70} a^{7} + \frac{3}{14} a^{6} - \frac{1}{14} a^{5} - \frac{8}{35} a^{4} + \frac{1}{10} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5880} a^{14} - \frac{1}{840} a^{13} - \frac{1}{420} a^{12} - \frac{1}{210} a^{10} + \frac{1}{60} a^{9} - \frac{1}{210} a^{8} + \frac{89}{1470} a^{7} - \frac{29}{210} a^{6} - \frac{1}{210} a^{5} + \frac{38}{105} a^{4} + \frac{1}{10} a^{3} - \frac{2}{15} a^{2} + \frac{1}{15} a - \frac{2}{15}$, $\frac{1}{11760} a^{15} - \frac{1}{840} a^{12} + \frac{13}{840} a^{11} - \frac{1}{840} a^{10} - \frac{1}{84} a^{9} + \frac{61}{2940} a^{8} - \frac{1}{10} a^{7} - \frac{6}{35} a^{6} - \frac{19}{140} a^{5} - \frac{2}{105} a^{4} - \frac{11}{30} a^{3} - \frac{13}{30} a^{2} - \frac{7}{30} a - \frac{1}{6}$, $\frac{1}{11760} a^{16} - \frac{1}{840} a^{13} + \frac{1}{840} a^{12} + \frac{1}{168} a^{11} + \frac{1}{60} a^{10} - \frac{11}{735} a^{9} - \frac{1}{70} a^{8} - \frac{8}{35} a^{7} - \frac{1}{20} a^{6} + \frac{1}{105} a^{5} - \frac{2}{21} a^{4} + \frac{1}{6} a^{3} - \frac{13}{30} a^{2} + \frac{1}{30} a + \frac{1}{5}$, $\frac{1}{23520} a^{17} - \frac{1}{11760} a^{14} + \frac{1}{1680} a^{13} - \frac{1}{1680} a^{12} + \frac{1}{84} a^{11} - \frac{43}{2940} a^{10} - \frac{1}{70} a^{9} - \frac{3}{140} a^{8} + \frac{307}{1960} a^{7} + \frac{1}{30} a^{6} - \frac{29}{420} a^{5} + \frac{59}{420} a^{4} - \frac{13}{60} a^{3} - \frac{29}{60} a^{2} + \frac{3}{10} a - \frac{1}{10}$, $\frac{1}{11041168248865440} a^{18} + \frac{179497014983}{11041168248865440} a^{17} - \frac{4022167057}{122679647209616} a^{16} - \frac{20861711119}{788654874918960} a^{15} + \frac{34360130801}{690073015554090} a^{14} - \frac{156052418957}{394327437459480} a^{13} - \frac{181068668119}{112664982131280} a^{12} + \frac{1475581853633}{306699118024040} a^{11} + \frac{2364048623065}{276029206221636} a^{10} + \frac{4443399845909}{276029206221636} a^{9} + \frac{4447940817641}{394327437459480} a^{8} - \frac{11287104144581}{306699118024040} a^{7} + \frac{4247417762017}{19716371872974} a^{6} + \frac{810766696337}{19716371872974} a^{5} + \frac{3581029130227}{98581859364870} a^{4} - \frac{35769287948}{1408312276641} a^{3} - \frac{581304629021}{9388748510940} a^{2} - \frac{2165350441387}{14083122766410} a - \frac{5870664137719}{14083122766410}$, $\frac{1}{390639491500192311355281528042274328222627963203777874526930613191802414664653275611229187807666329975888928654572863114864086741823667996580966560} a^{19} + \frac{3301992884166845002516567344348241450787697328212799090662965172940315448614792205305124835490861934753174665496644514781916889}{27902820821442307953948680574448166301616283085984133894780758085128743904618091115087799129119023569706352046755204508204577624415976285470069040} a^{18} + \frac{401662062921142510096765791987350920694236835602261656146200041579701012785455508415085082827132586484187843511821825414152542916691955287623}{78127898300038462271056305608454865644525592640755574905386122638360482932930655122245837561533265995177785730914572622972817348364733599316193312} a^{17} + \frac{328536891976324330410097224041961217935310890571199708769074657791669840740429761446415770918812458515330397636403581088839348579219172718487}{195319745750096155677640764021137164111313981601888937263465306595901207332326637805614593903833164987944464327286431557432043370911833998290483280} a^{16} + \frac{699471568932031066896581783570758172523385894370105877892063762180425340197989733403521817385327630071672259501127954923126290453393902403373}{27902820821442307953948680574448166301616283085984133894780758085128743904618091115087799129119023569706352046755204508204577624415976285470069040} a^{15} - \frac{1446584692851468496249352306185265571866545068502596910099989936699033048576419703459821455440843105451129885418177174936289509345178469257529}{24414968218762019459705095502642145513914247700236117157933163324487650916540829725701824237979145623493058040910803944679005421363979249786310410} a^{14} - \frac{14912933738838958191127049208186092500774199986234770120033851978770394793781199875122006243374113079655927381865796038844549295604800656797879}{9300940273814102651316226858149388767205427695328044631593586028376247968206030371695933043039674523235450682251734836068192541471992095156689680} a^{13} - \frac{68868406539173281145759952942151386525981305189679885715960703496802864719034652483951216292116745043412988211461085208154055590624464820142801}{39063949150019231135528152804227432822262796320377787452693061319180241466465327561122918780766632997588892865457286311486408674182366799658096656} a^{12} - \frac{113609831751763846460684552562273650571422227014512591970808149624030882603053406349981637345226259788435977489598460517315561569822189331648719}{6975705205360576988487170143612041575404070771496033473695189521282185976154522778771949782279755892426588011688801127051144406103994071367517260} a^{11} + \frac{536191724656246276539017383168272303958317823590948273554662605265293661663961150279865664468725899074696450677251492240426423312465307197512241}{48829936437524038919410191005284291027828495400472234315866326648975301833081659451403648475958291246986116081821607889358010842727958499572620820} a^{10} - \frac{16735905947786811572023516831383119856715147138417267902497703435438797081585584507060362180901355271031790951609970162033916676881938110180627}{2170219397223290618640452933568190712347933128909877080705170073287791192581407086729051043375924055421605159192071461749244926343464822203227592} a^{9} - \frac{1600904027604387756667773569470983983346297431128406215605571064604009701497978209125980047207674907270988518862663171556601258963910477134389}{348785260268028849424358507180602078770203538574801673684759476064109298807726138938597489113987794621329400584440056352557220305199703568375863} a^{8} + \frac{4428239818108275390938463093487214015880816066726984511763379746375544822869300377739177576522454311717055637110466855672322140780845299338043909}{97659872875048077838820382010568582055656990800944468631732653297950603666163318902807296951916582493972232163643215778716021685455916999145241640} a^{7} - \frac{771781479301141755511437183012497919661729788817860097426128793142779491173273607387909907322590899364080536141331196783966274868568681725731879}{6975705205360576988487170143612041575404070771496033473695189521282185976154522778771949782279755892426588011688801127051144406103994071367517260} a^{6} - \frac{246135307676458359663853195334231132987233343282502027942292146027488534051600832721822698838817617946194866048390978178767401061167112546259081}{1162617534226762831414528357268673595900678461916005578949198253547030996025753796461991630379959315404431335281466854508524067683999011894586210} a^{5} + \frac{24887609454392953267525010161862740965706061359097087038958022320812628012211270940164368323072689450187973512416931147895989944190276468175446}{116261753422676283141452835726867359590067846191600557894919825354703099602575379646199163037995931540443133528146685450852406768399901189458621} a^{4} + \frac{275194043644155223795435337221971570897705547881961492763049745592869240676195771325718966050677974245623403450142246985818247889655933312104787}{996529315051510998355310020516005939343438681642290496242169931611740853736360396967421397468536556060941144526971589578734915157713438766788180} a^{3} + \frac{50036431521357725928534298790318727989888865276354689355834353447543834659966593434022106815673345321989655457139650342270228657287095921801463}{996529315051510998355310020516005939343438681642290496242169931611740853736360396967421397468536556060941144526971589578734915157713438766788180} a^{2} - \frac{9263711089839485706566080057142849638135314815160394737174394021638123991515019907002699394444764356443138014137541265845383724608695525500191}{55362739725083944353072778917555885519079926757905027569009440645096714096464466498190077637140919781163396918165088309929717508761857709266010} a - \frac{134689097305735521416530683125842252915094032762340136911656757057787160314618958123232737143643531476358707852261838828332125650108114346290373}{498264657525755499177655010258002969671719340821145248121084965805870426868180198483710698734268278030470572263485794789367457578856719383394090}$

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

$D_4\times F_5$ (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

\(\Q(\sqrt{385}) \), 4.0.450604000.4, 5.1.5694792642000.6, 10.2.12485805345620275963140000000.1

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 R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.10.19.1$x^{10} - 2$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
$7$7.10.9.2$x^{10} + 14$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
7.10.9.2$x^{10} + 14$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$11$11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$