↑

Properties

Label 20.4.69817271402...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{22}\cdot 3^{17}\cdot 5^{15}\cdot 7^{18}\cdot 11^{10}$
Root discriminant $348.50$
Ramified primes $2, 3, 5, 7, 11$
Class number $160$ (GRH)
Class group $[2, 4, 20]$ (GRH)
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![-2101842107004, 466959547656, 795301311816, 47805685956, -122053122444, 52611971412, 25334742324, 4471295556, 720754398, 455452620, -90961241, -38299128, -5195701, -2265900, -218240, -26292, -2738, 0, 65, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 65*x^18 - 2738*x^16 - 26292*x^15 - 218240*x^14 - 2265900*x^13 - 5195701*x^12 - 38299128*x^11 - 90961241*x^10 + 455452620*x^9 + 720754398*x^8 + 4471295556*x^7 + 25334742324*x^6 + 52611971412*x^5 - 122053122444*x^4 + 47805685956*x^3 + 795301311816*x^2 + 466959547656*x - 2101842107004)
 
gp: K = bnfinit(x^20 + 65*x^18 - 2738*x^16 - 26292*x^15 - 218240*x^14 - 2265900*x^13 - 5195701*x^12 - 38299128*x^11 - 90961241*x^10 + 455452620*x^9 + 720754398*x^8 + 4471295556*x^7 + 25334742324*x^6 + 52611971412*x^5 - 122053122444*x^4 + 47805685956*x^3 + 795301311816*x^2 + 466959547656*x - 2101842107004, 1)
 

Normalized defining polynomial

\( x^{20} + 65 x^{18} - 2738 x^{16} - 26292 x^{15} - 218240 x^{14} - 2265900 x^{13} - 5195701 x^{12} - 38299128 x^{11} - 90961241 x^{10} + 455452620 x^{9} + 720754398 x^{8} + 4471295556 x^{7} + 25334742324 x^{6} + 52611971412 x^{5} - 122053122444 x^{4} + 47805685956 x^{3} + 795301311816 x^{2} + 466959547656 x - 2101842107004 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(698172714027048240382400334267313536000000000000000=2^{22}\cdot 3^{17}\cdot 5^{15}\cdot 7^{18}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $348.50$
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:  $2, 3, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{22} a^{12} - \frac{2}{11} a^{10} - \frac{7}{22} a^{9} - \frac{1}{2} a^{8} - \frac{1}{11} a^{7} + \frac{4}{11} a^{6} - \frac{9}{22} a^{5} + \frac{1}{11} a^{4} + \frac{1}{11} a^{3} + \frac{3}{11} a^{2} + \frac{5}{11} a + \frac{2}{11}$, $\frac{1}{22} a^{13} - \frac{2}{11} a^{11} + \frac{2}{11} a^{10} - \frac{1}{11} a^{8} + \frac{4}{11} a^{7} + \frac{1}{11} a^{6} - \frac{9}{22} a^{5} + \frac{1}{11} a^{4} + \frac{3}{11} a^{3} + \frac{5}{11} a^{2} + \frac{2}{11} a$, $\frac{1}{22} a^{14} + \frac{2}{11} a^{11} - \frac{5}{22} a^{10} + \frac{3}{22} a^{9} + \frac{4}{11} a^{8} - \frac{3}{11} a^{7} - \frac{5}{11} a^{6} - \frac{1}{22} a^{5} - \frac{4}{11} a^{4} - \frac{2}{11} a^{3} + \frac{3}{11} a^{2} - \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{22} a^{15} - \frac{5}{22} a^{11} - \frac{3}{22} a^{10} - \frac{4}{11} a^{9} - \frac{3}{11} a^{8} - \frac{1}{11} a^{7} - \frac{1}{2} a^{6} + \frac{3}{11} a^{5} + \frac{5}{11} a^{4} - \frac{1}{11} a^{3} - \frac{3}{11} a^{2} - \frac{1}{11} a + \frac{3}{11}$, $\frac{1}{66} a^{16} - \frac{1}{22} a^{11} + \frac{8}{33} a^{10} + \frac{1}{22} a^{9} + \frac{3}{22} a^{8} - \frac{7}{22} a^{7} + \frac{4}{11} a^{6} + \frac{3}{22} a^{5} + \frac{5}{11} a^{4} - \frac{3}{11} a^{3} + \frac{1}{11} a^{2} + \frac{2}{11} a - \frac{4}{11}$, $\frac{1}{1452} a^{17} - \frac{1}{726} a^{16} + \frac{1}{484} a^{15} + \frac{3}{242} a^{14} - \frac{1}{242} a^{13} - \frac{1}{242} a^{12} - \frac{65}{363} a^{11} + \frac{161}{726} a^{10} - \frac{235}{484} a^{9} + \frac{157}{484} a^{7} + \frac{7}{242} a^{6} + \frac{43}{242} a^{5} + \frac{21}{121} a^{4} - \frac{13}{242} a^{3} - \frac{97}{242} a^{2} + \frac{51}{121} a - \frac{54}{121}$, $\frac{1}{1811005548} a^{18} - \frac{74469}{603668516} a^{17} + \frac{2655449}{603668516} a^{16} - \frac{753779}{54878956} a^{15} - \frac{3275485}{150917129} a^{14} - \frac{847795}{150917129} a^{13} - \frac{1496431}{905502774} a^{12} - \frac{31975491}{301834258} a^{11} + \frac{106938459}{603668516} a^{10} + \frac{268970275}{603668516} a^{9} + \frac{260140323}{603668516} a^{8} + \frac{270809711}{603668516} a^{7} - \frac{29547556}{150917129} a^{6} - \frac{144261619}{301834258} a^{5} - \frac{119424097}{301834258} a^{4} - \frac{55460145}{150917129} a^{3} - \frac{90743717}{301834258} a^{2} + \frac{31261728}{150917129} a + \frac{5426976}{150917129}$, $\frac{1}{50879904133848838965637418575546261969180087706326302800729704778885393923904641907431419827176356600156} a^{19} - \frac{8272124192123786711543161216427181727406618694006515469736165696742136810361620886532229816085}{50879904133848838965637418575546261969180087706326302800729704778885393923904641907431419827176356600156} a^{18} - \frac{893445532230203208814982867663577357202489379676767105766560790934539112448318446670055852604179731}{50879904133848838965637418575546261969180087706326302800729704778885393923904641907431419827176356600156} a^{17} + \frac{329342875665694552127495325934579895564015755189530801932643041968835629904303877148228402066411228727}{50879904133848838965637418575546261969180087706326302800729704778885393923904641907431419827176356600156} a^{16} - \frac{57642518236374734291641495000808256818216497246577237641539723373657974539589210580390840180726816469}{8479984022308139827606236429257710328196681284387717133454950796480898987317440317905236637862726100026} a^{15} + \frac{60206588011641230298948465758539767012281198221749690879365892931089530427871807919785888454143257541}{4239992011154069913803118214628855164098340642193858566727475398240449493658720158952618318931363050013} a^{14} - \frac{4947679679089389305167942204839844400399282459655798535325784825357897632344700618985243544840986340}{1156361457587473612855395876716960499299547447871052336380220563156486225543287316077986814254008104549} a^{13} + \frac{360742147266277044819432932328016356632573672570349673461317595600505092488017799202278767844197368025}{25439952066924419482818709287773130984590043853163151400364852389442696961952320953715709913588178300078} a^{12} - \frac{11199373440572295759420520249290463528236352163898214887589925822656557431555046755136139792507753683059}{50879904133848838965637418575546261969180087706326302800729704778885393923904641907431419827176356600156} a^{11} - \frac{880541513288222303992824483250076752025558743431984608527862180496280729668004840660239529370971730189}{4625445830349894451421583506867841997198189791484209345520882252625944902173149264311947257016032418196} a^{10} + \frac{7249636603668693141581608927398619596979612748234819473106608276137381327663588845555222027964866485463}{16959968044616279655212472858515420656393362568775434266909901592961797974634880635810473275725452200052} a^{9} + \frac{6641007978301160968457138275554982909571183270713720533279955595906981007833403629158413840000236561273}{16959968044616279655212472858515420656393362568775434266909901592961797974634880635810473275725452200052} a^{8} - \frac{272929918971478027905897953472266446292767008143214243552707636899746273027693898720247368740325660793}{770907638391649075236930584477973666199698298580701557586813708770990817028858210718657876169338736366} a^{7} + \frac{1484056621808942434992674287182072111702074993927819194401522404283982172456914872383678655591505303715}{4239992011154069913803118214628855164098340642193858566727475398240449493658720158952618318931363050013} a^{6} + \frac{1143721900946332524469124136418596441890967080367682770628975483408045691814638119731672740637954317877}{8479984022308139827606236429257710328196681284387717133454950796480898987317440317905236637862726100026} a^{5} + \frac{1303117454546376785172753895091699851624581649019977396570479520553948604631416152179991049916908824616}{4239992011154069913803118214628855164098340642193858566727475398240449493658720158952618318931363050013} a^{4} - \frac{186628167233730729097327444105347473818895990022319165863351462057356905053058138166149437553247947527}{770907638391649075236930584477973666199698298580701557586813708770990817028858210718657876169338736366} a^{3} + \frac{823077440527780363941158144565496053573893190547837167486064920516334556062399353618005146916800576646}{4239992011154069913803118214628855164098340642193858566727475398240449493658720158952618318931363050013} a^{2} - \frac{592388289480571968091285289745532849918523928540107793600856416354860098962017534021125397461075056878}{4239992011154069913803118214628855164098340642193858566727475398240449493658720158952618318931363050013} a + \frac{355714963736007473935825630347283557559563659584599580851381049338222928431182016946522892049605486508}{4239992011154069913803118214628855164098340642193858566727475398240449493658720158952618318931363050013}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{20}$, which has order $160$ (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:  $11$
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:  \( 75775460004876270 \) (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

\(\Q(\sqrt{385}) \), 4.4.8893500.3, 5.1.388962000.4, 10.2.852797305212777540000000.1

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 ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{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$$[2]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{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$$[\ ]_{10}^{4}$
$11$11.2.1.2$x^{2} + 33$$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}$