↑

Properties

Label 20.2.49763101145...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{38}\cdot 3^{18}\cdot 5^{37}\cdot 11^{10}\cdot 19^{5}$
Root discriminant $1364.10$
Ramified primes $2, 3, 5, 11, 19$
Class number Not computed
Class group Not computed
Galois group $D_4\times F_5$ (as 20T42)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-79045938974448, -3991415434560, 7262365534560, 410323462560, 414207451440, -233407352952, -78897940920, 22947160680, 3154256910, -132271680, -95388407, -70749730, 2200905, 6684000, -492270, -161532, 14520, 2040, -195, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 195*x^18 + 2040*x^17 + 14520*x^16 - 161532*x^15 - 492270*x^14 + 6684000*x^13 + 2200905*x^12 - 70749730*x^11 - 95388407*x^10 - 132271680*x^9 + 3154256910*x^8 + 22947160680*x^7 - 78897940920*x^6 - 233407352952*x^5 + 414207451440*x^4 + 410323462560*x^3 + 7262365534560*x^2 - 3991415434560*x - 79045938974448)
 
gp: K = bnfinit(x^20 - 10*x^19 - 195*x^18 + 2040*x^17 + 14520*x^16 - 161532*x^15 - 492270*x^14 + 6684000*x^13 + 2200905*x^12 - 70749730*x^11 - 95388407*x^10 - 132271680*x^9 + 3154256910*x^8 + 22947160680*x^7 - 78897940920*x^6 - 233407352952*x^5 + 414207451440*x^4 + 410323462560*x^3 + 7262365534560*x^2 - 3991415434560*x - 79045938974448, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 195 x^{18} + 2040 x^{17} + 14520 x^{16} - 161532 x^{15} - 492270 x^{14} + 6684000 x^{13} + 2200905 x^{12} - 70749730 x^{11} - 95388407 x^{10} - 132271680 x^{9} + 3154256910 x^{8} + 22947160680 x^{7} - 78897940920 x^{6} - 233407352952 x^{5} + 414207451440 x^{4} + 410323462560 x^{3} + 7262365534560 x^{2} - 3991415434560 x - 79045938974448 \)

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:  \(-497631011454939064202060220000000000000000000000000000000000000=-\,2^{38}\cdot 3^{18}\cdot 5^{37}\cdot 11^{10}\cdot 19^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1364.10$
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:  $2, 3, 5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{18} a^{8} - \frac{1}{18} a^{7} + \frac{1}{9} a^{5} + \frac{1}{18} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{18} a^{9} - \frac{1}{18} a^{7} - \frac{1}{18} a^{6} + \frac{1}{18} a^{4}$, $\frac{1}{108} a^{10} + \frac{1}{108} a^{9} - \frac{1}{54} a^{8} - \frac{1}{27} a^{7} - \frac{1}{108} a^{6} + \frac{11}{108} a^{5} - \frac{1}{18} a^{4}$, $\frac{1}{108} a^{11} - \frac{1}{36} a^{9} - \frac{1}{54} a^{8} + \frac{1}{36} a^{7} - \frac{1}{18} a^{6} + \frac{1}{108} a^{5} - \frac{1}{9} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{324} a^{12} + \frac{1}{324} a^{9} - \frac{1}{108} a^{8} - \frac{5}{81} a^{6} + \frac{13}{108} a^{5} - \frac{1}{18} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{324} a^{13} + \frac{1}{324} a^{10} - \frac{1}{108} a^{9} - \frac{5}{81} a^{7} - \frac{5}{108} a^{6} + \frac{1}{9} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3}$, $\frac{1}{972} a^{14} - \frac{1}{972} a^{13} + \frac{1}{972} a^{11} - \frac{1}{243} a^{10} + \frac{1}{324} a^{9} - \frac{5}{243} a^{8} + \frac{59}{972} a^{7} - \frac{19}{324} a^{6} + \frac{7}{54} a^{5} - \frac{1}{9} a^{4} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{24300} a^{15} + \frac{1}{4860} a^{14} + \frac{1}{1620} a^{13} + \frac{1}{2430} a^{12} + \frac{13}{4860} a^{11} - \frac{1}{450} a^{10} - \frac{103}{4860} a^{9} + \frac{19}{1215} a^{8} - \frac{59}{810} a^{7} + \frac{43}{540} a^{6} + \frac{167}{1350} a^{5} + \frac{1}{45} a^{4} + \frac{13}{90} a^{3} + \frac{7}{15} a^{2} - \frac{2}{5} a + \frac{3}{25}$, $\frac{1}{534600} a^{16} + \frac{1}{53460} a^{15} + \frac{7}{26730} a^{14} + \frac{7}{17820} a^{13} + \frac{19}{53460} a^{12} + \frac{73}{267300} a^{11} - \frac{31}{8910} a^{10} - \frac{947}{53460} a^{9} + \frac{301}{106920} a^{8} - \frac{436}{13365} a^{7} - \frac{127}{7425} a^{6} + \frac{19}{2970} a^{5} - \frac{1}{165} a^{4} + \frac{131}{990} a^{3} - \frac{134}{495} a^{2} + \frac{217}{825} a - \frac{1}{55}$, $\frac{1}{534600} a^{17} - \frac{1}{133650} a^{15} + \frac{2}{4455} a^{14} + \frac{73}{53460} a^{13} - \frac{68}{66825} a^{12} + \frac{7}{8910} a^{11} - \frac{1}{12150} a^{10} - \frac{647}{106920} a^{9} + \frac{296}{13365} a^{8} + \frac{7159}{133650} a^{7} - \frac{553}{8910} a^{6} + \frac{406}{2475} a^{5} + \frac{46}{495} a^{4} + \frac{281}{990} a^{3} + \frac{916}{2475} a^{2} - \frac{2}{11} a - \frac{16}{275}$, $\frac{1}{16029292780084887051219748549800} a^{18} + \frac{251105746637813937128087}{485736144851057183370295410600} a^{17} + \frac{21001922710087190885812}{74209688796689291903795132175} a^{16} - \frac{138114740026283354589527}{6746335345153571991254102925} a^{15} - \frac{54512251926307934147727157}{178103253112054300569108317220} a^{14} - \frac{364186786476650299517706127}{1335774398340407254268312379150} a^{13} - \frac{405882894922375086886606237}{1335774398340407254268312379150} a^{12} + \frac{193225622504009516365340981}{98946251728919055871726842900} a^{11} + \frac{5865693749765525443703480351}{5343097593361629017073249516600} a^{10} - \frac{62330769004508226754774878299}{3205858556016977410243949709960} a^{9} - \frac{8677235056036692168044221466}{667887199170203627134156189575} a^{8} + \frac{150510246998295750239231430841}{2671548796680814508536624758300} a^{7} - \frac{1794279988545539930154227881}{24736562932229763967931710725} a^{6} - \frac{1509669918331732198911411637}{74209688796689291903795132175} a^{5} + \frac{1642525103588854832995454002}{14841937759337858380759026435} a^{4} + \frac{1744457138523960414183335641}{5497013984939947548429269050} a^{3} + \frac{3950832298908802937837323607}{24736562932229763967931710725} a^{2} + \frac{3661587108468620470275890443}{8245520977409921322643903575} a - \frac{88841084406889896862585006}{2748506992469973774214634525}$, $\frac{1}{6982593285253802439749014963669224105838561527426269927169000} a^{19} + \frac{10798374778872154770544903381}{2327531095084600813249671654556408035279520509142089975723000} a^{18} + \frac{687001201245093236211888615713366844533721336985253179}{1163765547542300406624835827278204017639760254571044987861500} a^{17} + \frac{791195274728935731872780644833413889938557136306031067}{1163765547542300406624835827278204017639760254571044987861500} a^{16} + \frac{2928286585847237103115972523800972101311173706908167911}{1163765547542300406624835827278204017639760254571044987861500} a^{15} - \frac{136469974339300306742393069092447082983424772052343564213}{387921849180766802208278609092734672546586751523681662620500} a^{14} - \frac{43408773539344497996467212548734012242889731652164411859}{64653641530127800368046434848789112091097791920613610436750} a^{13} + \frac{1500044448771578868901167312927710096963842029828065248679}{1163765547542300406624835827278204017639760254571044987861500} a^{12} + \frac{1981226797221847895519232195974306874156079917279771400763}{775843698361533604416557218185469345093173503047363325241000} a^{11} + \frac{4738975951921912735174192453088219793550894994027115958211}{6982593285253802439749014963669224105838561527426269927169000} a^{10} - \frac{9862755462394763161611546186003116671707654628779235524493}{387921849180766802208278609092734672546586751523681662620500} a^{9} - \frac{104536331692127207382790530391873666710426638932990415098}{290941386885575101656208956819551004409940063642761246965375} a^{8} + \frac{53309918783823539662907581447397294435225129103917463093499}{1163765547542300406624835827278204017639760254571044987861500} a^{7} + \frac{6197710318521824794289895627899691640384039717063339832216}{96980462295191700552069652273183668136646687880920415655125} a^{6} - \frac{10270972106838412055357226155805372246891463933977622480103}{64653641530127800368046434848789112091097791920613610436750} a^{5} + \frac{157346249205950799916903485277226178220803222885916689959}{1959201258488721223274134389357245820942357330927685164750} a^{4} - \frac{157413703030659757734318701306098106936381869381246494884}{10775606921687966728007739141464852015182965320102268406125} a^{3} - \frac{3097771584308689125140254445200202884307676786961348907302}{10775606921687966728007739141464852015182965320102268406125} a^{2} - \frac{1214425963163750107299713705295903745897744234193897516602}{3591868973895988909335913047154950671727655106700756135375} a + \frac{370704629752596373114292810229583451394595456590275858913}{1197289657965329636445304349051650223909218368900252045125}$

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

\(\Q(\sqrt{66}) \), 4.2.6621120.8, 5.1.2531250000.13, 10.2.6339933666000000000000000000.1

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 ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.10.19.1$x^{10} - 2$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.19.1$x^{10} - 2$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.3$x^{10} + 30$$10$$1$$19$$F_5$$[9/4]_{4}$
$11$11.2.1.2$x^{2} + 33$$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$$[\ ]_{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$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}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$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}$