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Properties

Label 20.2.49941662158...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{28}\cdot 3^{18}\cdot 5^{38}\cdot 11^{5}\cdot 31^{10}$
Root discriminant $1530.82$
Ramified primes $2, 3, 5, 11, 31$
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![4796675607065476, -3858203116950000, 1214937460911860, -573879860100000, 87823626162120, -43635003577500, -2795604863880, -2089986637500, -7942171740, -24311775000, -9341523, -840412500, -14586315, 1387500, -14430, -15000, 10170, 0, 185, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 185*x^18 + 10170*x^16 - 15000*x^15 - 14430*x^14 + 1387500*x^13 - 14586315*x^12 - 840412500*x^11 - 9341523*x^10 - 24311775000*x^9 - 7942171740*x^8 - 2089986637500*x^7 - 2795604863880*x^6 - 43635003577500*x^5 + 87823626162120*x^4 - 573879860100000*x^3 + 1214937460911860*x^2 - 3858203116950000*x + 4796675607065476)
 
gp: K = bnfinit(x^20 + 185*x^18 + 10170*x^16 - 15000*x^15 - 14430*x^14 + 1387500*x^13 - 14586315*x^12 - 840412500*x^11 - 9341523*x^10 - 24311775000*x^9 - 7942171740*x^8 - 2089986637500*x^7 - 2795604863880*x^6 - 43635003577500*x^5 + 87823626162120*x^4 - 573879860100000*x^3 + 1214937460911860*x^2 - 3858203116950000*x + 4796675607065476, 1)
 

Normalized defining polynomial

\( x^{20} + 185 x^{18} + 10170 x^{16} - 15000 x^{15} - 14430 x^{14} + 1387500 x^{13} - 14586315 x^{12} - 840412500 x^{11} - 9341523 x^{10} - 24311775000 x^{9} - 7942171740 x^{8} - 2089986637500 x^{7} - 2795604863880 x^{6} - 43635003577500 x^{5} + 87823626162120 x^{4} - 573879860100000 x^{3} + 1214937460911860 x^{2} - 3858203116950000 x + 4796675607065476 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4994166215876227776290092298730468750000000000000000000000000000=-\,2^{28}\cdot 3^{18}\cdot 5^{38}\cdot 11^{5}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1530.82$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{2} - \frac{2}{5}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{25} a^{8} - \frac{1}{25} a^{6} + \frac{1}{25} a^{4} + \frac{9}{25} a^{2} + \frac{6}{25}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{7} + \frac{1}{25} a^{5} + \frac{9}{25} a^{3} + \frac{6}{25} a$, $\frac{1}{2250} a^{10} - \frac{1}{50} a^{9} - \frac{2}{225} a^{8} - \frac{2}{25} a^{7} + \frac{41}{450} a^{6} + \frac{7}{150} a^{5} + \frac{17}{225} a^{4} - \frac{31}{75} a^{3} + \frac{74}{225} a^{2} - \frac{14}{75} a + \frac{523}{1125}$, $\frac{1}{2250} a^{11} + \frac{1}{90} a^{9} - \frac{13}{450} a^{7} + \frac{1}{15} a^{6} + \frac{43}{450} a^{5} + \frac{1}{15} a^{4} - \frac{43}{225} a^{3} - \frac{1}{15} a^{2} - \frac{242}{1125} a - \frac{2}{5}$, $\frac{1}{2250} a^{12} - \frac{1}{50} a^{9} - \frac{1}{150} a^{8} - \frac{1}{75} a^{7} + \frac{4}{225} a^{6} - \frac{1}{50} a^{5} - \frac{2}{25} a^{4} - \frac{31}{75} a^{3} + \frac{61}{375} a^{2} - \frac{4}{75} a + \frac{17}{45}$, $\frac{1}{2250} a^{13} + \frac{1}{75} a^{9} - \frac{1}{75} a^{8} + \frac{22}{225} a^{7} + \frac{2}{25} a^{6} - \frac{3}{50} a^{5} - \frac{1}{75} a^{4} + \frac{16}{375} a^{3} + \frac{26}{75} a^{2} + \frac{67}{225} a + \frac{3}{25}$, $\frac{1}{22500} a^{14} - \frac{1}{4500} a^{13} + \frac{1}{11250} a^{12} - \frac{1}{5625} a^{10} - \frac{1}{50} a^{9} - \frac{4}{375} a^{8} - \frac{8}{225} a^{7} - \frac{73}{4500} a^{6} + \frac{1}{20} a^{5} - \frac{551}{5625} a^{4} - \frac{51}{125} a^{3} - \frac{187}{5625} a^{2} - \frac{13}{450} a + \frac{221}{3750}$, $\frac{1}{90000} a^{15} + \frac{17}{90000} a^{13} - \frac{7}{45000} a^{11} - \frac{1}{9000} a^{10} + \frac{161}{9000} a^{9} - \frac{11}{1800} a^{8} + \frac{179}{6000} a^{7} - \frac{7}{225} a^{6} - \frac{8479}{90000} a^{5} - \frac{77}{900} a^{4} + \frac{679}{1875} a^{3} + \frac{497}{1800} a^{2} - \frac{4423}{11250} a - \frac{2471}{9000}$, $\frac{1}{450000} a^{16} - \frac{7}{450000} a^{14} - \frac{31}{225000} a^{12} - \frac{1}{9000} a^{11} - \frac{47}{225000} a^{10} + \frac{1}{1800} a^{9} + \frac{83}{30000} a^{8} + \frac{43}{450} a^{7} + \frac{28481}{450000} a^{6} + \frac{79}{900} a^{5} - \frac{248}{3125} a^{4} + \frac{97}{360} a^{3} + \frac{10721}{56250} a^{2} + \frac{2989}{9000} a - \frac{721}{3125}$, $\frac{1}{450000} a^{17} - \frac{1}{225000} a^{15} + \frac{23}{450000} a^{13} - \frac{1}{9000} a^{12} + \frac{1}{12500} a^{11} + \frac{353}{30000} a^{9} + \frac{11}{600} a^{8} - \frac{3547}{225000} a^{7} - \frac{79}{900} a^{6} + \frac{15893}{450000} a^{5} + \frac{19}{200} a^{4} + \frac{23341}{56250} a^{3} + \frac{139}{1500} a^{2} - \frac{1827}{6250} a - \frac{79}{360}$, $\frac{1}{163776168494867672742401599314150000} a^{18} + \frac{19214751125095617710310700763}{27296028082477945457066933219025000} a^{17} + \frac{7489727621115846354263108101}{9098676027492648485688977739675000} a^{16} - \frac{76900276313225994854136190121}{27296028082477945457066933219025000} a^{15} + \frac{28983307465976458247179193369}{6065784018328432323792651826450000} a^{14} + \frac{4696820282408435316746354744459}{27296028082477945457066933219025000} a^{13} + \frac{82879707708567785234722866949}{1137334503436581060711122217459375} a^{12} - \frac{485911412409923628101901983767}{4549338013746324242844488869837500} a^{11} - \frac{601245073213282483239035802787}{10918411232991178182826773287610000} a^{10} - \frac{1814076787772842482411130250483}{909867602749264848568897773967500} a^{9} - \frac{139465279703846728145483253260899}{27296028082477945457066933219025000} a^{8} + \frac{2128290249802481937533230401297953}{27296028082477945457066933219025000} a^{7} + \frac{368378771443844462256870373174819}{6065784018328432323792651826450000} a^{6} - \frac{338570025088598137955786831656511}{27296028082477945457066933219025000} a^{5} - \frac{143297718883151930712952990481611}{13648014041238972728533466609512500} a^{4} - \frac{1906278558315080321487079273536563}{13648014041238972728533466609512500} a^{3} + \frac{933442901201103366770671362852473}{13648014041238972728533466609512500} a^{2} - \frac{10103862190846122079899662520599617}{27296028082477945457066933219025000} a - \frac{110346979963576653680403893912557}{409440421237169181856003998285375}$, $\frac{1}{826039339554040180791854432381513336841915025989144692466646755787224300000} a^{19} + \frac{33510083100390133281106455542169604527}{15297024806556299644293600599657654385961389370169346156789754736800450000} a^{18} - \frac{12138655794908130858786439403454482691381016594623379993819962856907}{11013857860720535743891392431753511157892200346521929232888623410496324000} a^{17} - \frac{7027209122975271050858370378722242325075382534161157527193490151513}{17209152907375837099830300674614861184206563041440514426388474078900506250} a^{16} + \frac{107171930013548223492056184026192847306657484430489363547203375891971}{34418305814751674199660601349229722368413126082881028852776948157801012500} a^{15} - \frac{13372264038263619928056833322712207187206238286280425257546920491751}{1376732232590066967986424053969188894736525043315241154111077926312040500} a^{14} - \frac{27708576589269909827507478929265201637810876046249588740452765110030233}{137673223259006696798642405396918889473652504331524115411107792631204050000} a^{13} + \frac{19417031449792601260908243763311308613972662921571659176200134714229409}{137673223259006696798642405396918889473652504331524115411107792631204050000} a^{12} - \frac{1110922894972865585629247853356859340515571681974962819452116342340847}{91782148839337797865761603597945926315768336221016076940738528420802700000} a^{11} + \frac{19527885438891872881726358306521319255176809056647042305070934934641387}{137673223259006696798642405396918889473652504331524115411107792631204050000} a^{10} - \frac{3114668953853413686882700184005477891918177822606437845402943946881073343}{275346446518013393597284810793837778947305008663048230822215585262408100000} a^{9} - \frac{20655568850975998868100052523608027229562351855680515175919593279171317}{22945537209834449466440400899486481578942084055254019235184632105200675000} a^{8} - \frac{649371410573297930563062803626588889282859020960808705385082931738212299}{27534644651801339359728481079383777894730500866304823082221558526240810000} a^{7} + \frac{539573001569671264029486258793321908457537618941989457642383536283140759}{5736384302458612366610100224871620394735521013813504808796158026300168750} a^{6} + \frac{5006603243534981252182273327352255845069994843017874071697327467537360427}{68836611629503348399321202698459444736826252165762057705553896315602025000} a^{5} - \frac{46744124925164064710997819251239154598603994947144132782374713150180151}{949470505234528943438913140668406134301051754010511140766260638835890000} a^{4} - \frac{848697658135434766270267709284530803762292616411619797210052833932001999}{1912128100819537455536700074957206798245173671271168269598719342100056250} a^{3} + \frac{23986814566671986217021230535520674338845194821398138686998199385547412077}{68836611629503348399321202698459444736826252165762057705553896315602025000} a^{2} - \frac{6572895665119729482898765869942324924611714867887263570292393800531950257}{206509834888510045197963608095378334210478756497286173116661688946806075000} a - \frac{1370645426247221324186488599424628547784040832972579717775490030559942538}{8604576453687918549915150337307430592103281520720257213194237039450253125}$

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:  $10$
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

\(\Q(\sqrt{465}) \), 4.2.38055600.3, 5.1.2531250000.1, 10.2.2751501851235351562500000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.10.19.19$x^{10} + 110$$10$$1$$19$$F_{5}\times C_2$$[9/4]_{4}^{2}$
5.10.19.19$x^{10} + 110$$10$$1$$19$$F_{5}\times C_2$$[9/4]_{4}^{2}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$