↑

Properties

Label 20.2.77902589415...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{16}\cdot 3^{16}\cdot 5^{38}\cdot 29^{10}\cdot 71^{5}$
Root discriminant $1395.01$
Ramified primes $2, 3, 5, 29, 71$
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![25851941971250, 4515903081250, 4635668881250, -1527571833750, 718698621250, -1676184575, 56145238125, -15837478375, 1251634225, 607794475, -30396651, 20182195, -10398055, 1207235, 15315, -33161, 2585, 735, -95, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 95*x^18 + 735*x^17 + 2585*x^16 - 33161*x^15 + 15315*x^14 + 1207235*x^13 - 10398055*x^12 + 20182195*x^11 - 30396651*x^10 + 607794475*x^9 + 1251634225*x^8 - 15837478375*x^7 + 56145238125*x^6 - 1676184575*x^5 + 718698621250*x^4 - 1527571833750*x^3 + 4635668881250*x^2 + 4515903081250*x + 25851941971250)
 
gp: K = bnfinit(x^20 - 5*x^19 - 95*x^18 + 735*x^17 + 2585*x^16 - 33161*x^15 + 15315*x^14 + 1207235*x^13 - 10398055*x^12 + 20182195*x^11 - 30396651*x^10 + 607794475*x^9 + 1251634225*x^8 - 15837478375*x^7 + 56145238125*x^6 - 1676184575*x^5 + 718698621250*x^4 - 1527571833750*x^3 + 4635668881250*x^2 + 4515903081250*x + 25851941971250, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 95 x^{18} + 735 x^{17} + 2585 x^{16} - 33161 x^{15} + 15315 x^{14} + 1207235 x^{13} - 10398055 x^{12} + 20182195 x^{11} - 30396651 x^{10} + 607794475 x^{9} + 1251634225 x^{8} - 15837478375 x^{7} + 56145238125 x^{6} - 1676184575 x^{5} + 718698621250 x^{4} - 1527571833750 x^{3} + 4635668881250 x^{2} + 4515903081250 x + 25851941971250 \)

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:  \(-779025894158619568967484436566138267517089843750000000000000000=-\,2^{16}\cdot 3^{16}\cdot 5^{38}\cdot 29^{10}\cdot 71^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1395.01$
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:  $2, 3, 5, 29, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3}$, $\frac{1}{15} a^{8} + \frac{1}{15} a^{7} + \frac{1}{15} a^{6} + \frac{7}{15} a^{5} + \frac{1}{5} a^{4} + \frac{2}{15} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{9} + \frac{1}{3} a^{5} + \frac{2}{15} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{150} a^{10} - \frac{1}{30} a^{9} + \frac{1}{30} a^{6} - \frac{23}{150} a^{5} + \frac{4}{15} a^{4} + \frac{2}{5} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{300} a^{11} + \frac{1}{60} a^{9} - \frac{1}{12} a^{7} - \frac{7}{75} a^{6} - \frac{9}{20} a^{5} - \frac{7}{30} a^{4} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{300} a^{12} - \frac{1}{300} a^{10} - \frac{1}{30} a^{9} - \frac{1}{60} a^{8} - \frac{2}{75} a^{7} - \frac{1}{12} a^{6} + \frac{32}{75} a^{5} - \frac{1}{15} a^{4} + \frac{3}{10} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{300} a^{13} - \frac{1}{30} a^{9} - \frac{2}{75} a^{8} + \frac{1}{30} a^{7} - \frac{1}{10} a^{6} - \frac{1}{60} a^{5} - \frac{1}{3} a^{4} - \frac{1}{15} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{1500} a^{14} - \frac{1}{1500} a^{13} - \frac{1}{1500} a^{12} - \frac{1}{300} a^{10} - \frac{7}{375} a^{9} + \frac{43}{1500} a^{8} + \frac{9}{125} a^{7} + \frac{3}{50} a^{6} + \frac{29}{300} a^{5} - \frac{1}{3} a^{4} - \frac{7}{30} a^{3} + \frac{1}{5} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6000} a^{15} + \frac{1}{2000} a^{13} + \frac{1}{1500} a^{12} - \frac{1}{600} a^{11} + \frac{1}{3000} a^{10} + \frac{3}{200} a^{9} - \frac{7}{3000} a^{8} - \frac{367}{6000} a^{7} - \frac{1}{15} a^{6} - \frac{277}{1200} a^{5} + \frac{11}{30} a^{4} - \frac{29}{120} a^{2} - \frac{1}{3} a + \frac{3}{8}$, $\frac{1}{18000} a^{16} - \frac{1}{18000} a^{15} + \frac{1}{6000} a^{14} + \frac{1}{18000} a^{13} + \frac{1}{3000} a^{12} + \frac{1}{1500} a^{11} + \frac{7}{4500} a^{10} - \frac{13}{2250} a^{9} - \frac{151}{6000} a^{8} - \frac{193}{18000} a^{7} - \frac{97}{3600} a^{6} - \frac{83}{400} a^{5} + \frac{7}{30} a^{4} + \frac{151}{360} a^{3} - \frac{17}{120} a^{2} + \frac{13}{72} a + \frac{31}{72}$, $\frac{1}{36000} a^{17} + \frac{1}{18000} a^{15} + \frac{1}{9000} a^{14} + \frac{7}{36000} a^{13} - \frac{7}{6000} a^{12} - \frac{1}{1800} a^{11} - \frac{1}{2250} a^{10} - \frac{257}{36000} a^{9} - \frac{173}{18000} a^{8} - \frac{383}{6000} a^{7} + \frac{16}{225} a^{6} + \frac{59}{2400} a^{5} - \frac{61}{144} a^{4} + \frac{4}{45} a^{3} - \frac{83}{360} a^{2} - \frac{1}{9} a - \frac{29}{144}$, $\frac{1}{180000} a^{18} - \frac{1}{90000} a^{17} + \frac{1}{45000} a^{16} - \frac{1}{22500} a^{15} - \frac{19}{180000} a^{14} + \frac{1}{2500} a^{13} - \frac{71}{45000} a^{12} + \frac{19}{15000} a^{11} - \frac{181}{180000} a^{10} + \frac{173}{45000} a^{9} + \frac{109}{9000} a^{8} - \frac{827}{9000} a^{7} - \frac{199}{2400} a^{6} - \frac{331}{1800} a^{5} - \frac{83}{1800} a^{4} + \frac{23}{90} a^{3} - \frac{1}{20} a^{2} + \frac{1}{144} a + \frac{11}{24}$, $\frac{1}{278727744627322930731937012300701443576148208835962238624340402533557468032155075114580000} a^{19} + \frac{369628717322649063609483367385387591792366030854503996409909257652411271687180188537}{278727744627322930731937012300701443576148208835962238624340402533557468032155075114580000} a^{18} + \frac{3723032957478052130268554750114660428030658893572957135613271865394227291286984732501}{278727744627322930731937012300701443576148208835962238624340402533557468032155075114580000} a^{17} - \frac{2682796711207023725878190737098698196292562696728483368194303801371582532967070848691}{139363872313661465365968506150350721788074104417981119312170201266778734016077537557290000} a^{16} - \frac{14979056696758472339094795422857006620780361404595931555397187966696903090863672296131}{278727744627322930731937012300701443576148208835962238624340402533557468032155075114580000} a^{15} - \frac{3383965511420080165701328493198803678702472840860260435325155820563057707982289550513}{92909248209107643577312337433567147858716069611987412874780134177852489344051691704860000} a^{14} + \frac{446004515918089064757948513262748234152000842911503169590311453002930654805424569804259}{278727744627322930731937012300701443576148208835962238624340402533557468032155075114580000} a^{13} + \frac{13430252831069247082624846010309795714622338864535140863776594300548717148605881504597}{46454624104553821788656168716783573929358034805993706437390067088926244672025845852430000} a^{12} - \frac{348008278831545066765877653680317008660554272464281725590123637333059957751788776176549}{278727744627322930731937012300701443576148208835962238624340402533557468032155075114580000} a^{11} - \frac{69283240399641504981425931688543419135178470894146826162186443272208287033840089478267}{278727744627322930731937012300701443576148208835962238624340402533557468032155075114580000} a^{10} + \frac{1031207114392441773837042560128215822238694550782110787327291872381767513102771832195553}{278727744627322930731937012300701443576148208835962238624340402533557468032155075114580000} a^{9} + \frac{79154604542299359129602165890397150042954802776185349343334154240949922569124698439331}{2787277446273229307319370123007014435761482088359622386243404025335574680321550751145800} a^{8} + \frac{286894804939663507977417522610072761794824006927739258415269084735311112810344856960929}{3716369928364305743092493497342685914348642784479496514991205367114099573762067668194400} a^{7} + \frac{769625671500903286471680084980703786962870021692385782279213021515271018565171092147511}{11149109785092917229277480492028057743045928353438489544973616101342298721286203004583200} a^{6} - \frac{4764700562282078016503764317751868527554304431798616860836690883898289365053243424100087}{11149109785092917229277480492028057743045928353438489544973616101342298721286203004583200} a^{5} - \frac{664108240372734442536955122049257108788487755050941837183318106235488499637154000162369}{5574554892546458614638740246014028871522964176719244772486808050671149360643101502291600} a^{4} - \frac{42452495942598882618257944672116274627229802931544954355810783456677704722491061687823}{185818496418215287154624674867134295717432139223974825749560268355704978688103383409720} a^{3} + \frac{41950668010583907713751545030682743265011560455831419323627814040239416956152627920705}{222982195701858344585549609840561154860918567068769790899472322026845974425724060091664} a^{2} - \frac{24646423479772443646495908288804092878305849066199921422334293073793601199561699465655}{74327398567286114861849869946853718286972855689589930299824107342281991475241353363888} a - \frac{1950081105292381554735704923184947937904804582760191213363804043474226218605333363559}{24775799522428704953949956648951239428990951896529976766608035780760663825080451121296}$

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

\(\Q(\sqrt{145}) \), 4.2.1492775.2, 5.1.2531250000.5, 10.2.657097893500976562500000000.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ R $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$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}$
$5$5.10.19.5$x^{10} + 55$$10$$1$$19$$F_5$$[9/4]_{4}$
5.10.19.5$x^{10} + 55$$10$$1$$19$$F_5$$[9/4]_{4}$
$29$29.4.2.1$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}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$71$71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$