Properties

Label 20.0.26762285357...3376.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 761^{5}$
Root discriminant $10.50$
Ramified primes $2, 761$
Class number $1$
Class group Trivial
Galois Group $D_5\wr C_2$ (as 20T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 27, -54, 88, -144, 188, -140, 44, -34, 92, -92, 25, 4, 17, -18, 6, 0, 0, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*x + 1)
gp: K = bnfinit(x^20 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 2 x^{19} \) \(\mathstrut +\mathstrut 6 x^{16} \) \(\mathstrut -\mathstrut 18 x^{15} \) \(\mathstrut +\mathstrut 17 x^{14} \) \(\mathstrut +\mathstrut 4 x^{13} \) \(\mathstrut +\mathstrut 25 x^{12} \) \(\mathstrut -\mathstrut 92 x^{11} \) \(\mathstrut +\mathstrut 92 x^{10} \) \(\mathstrut -\mathstrut 34 x^{9} \) \(\mathstrut +\mathstrut 44 x^{8} \) \(\mathstrut -\mathstrut 140 x^{7} \) \(\mathstrut +\mathstrut 188 x^{6} \) \(\mathstrut -\mathstrut 144 x^{5} \) \(\mathstrut +\mathstrut 88 x^{4} \) \(\mathstrut -\mathstrut 54 x^{3} \) \(\mathstrut +\mathstrut 27 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(267622853577068773376=2^{20}\cdot 761^{5}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.50$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 761$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{4379139123} a^{19} - \frac{119017966}{4379139123} a^{18} + \frac{1685921074}{4379139123} a^{17} - \frac{2114361461}{4379139123} a^{16} + \frac{33058990}{1459713041} a^{15} + \frac{365282}{257596419} a^{14} + \frac{114632567}{257596419} a^{13} - \frac{973911976}{4379139123} a^{12} - \frac{1115738719}{4379139123} a^{11} + \frac{537527623}{1459713041} a^{10} - \frac{379823704}{1459713041} a^{9} + \frac{493709918}{1459713041} a^{8} + \frac{230256856}{1459713041} a^{7} - \frac{1268264551}{4379139123} a^{6} + \frac{160458095}{4379139123} a^{5} + \frac{746923456}{4379139123} a^{4} - \frac{934160479}{4379139123} a^{3} - \frac{270645920}{4379139123} a^{2} + \frac{754334224}{4379139123} a + \frac{1554696913}{4379139123}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( -\frac{3778880}{270869} a^{19} + \frac{11227566}{270869} a^{18} - \frac{1330507}{270869} a^{17} - \frac{6019962}{270869} a^{16} - \frac{29575666}{270869} a^{15} + \frac{83845502}{270869} a^{14} - \frac{99234599}{270869} a^{13} - \frac{28701735}{270869} a^{12} - \frac{57842432}{270869} a^{11} + \frac{495643623}{270869} a^{10} - \frac{479848985}{270869} a^{9} + \frac{109683743}{270869} a^{8} - \frac{126499126}{270869} a^{7} + \frac{696261730}{270869} a^{6} - \frac{967858891}{270869} a^{5} + \frac{644490779}{270869} a^{4} - \frac{358982021}{270869} a^{3} + \frac{241607430}{270869} a^{2} - \frac{112651583}{270869} a + \frac{21204039}{270869} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{22484982706}{257596419} a^{19} - \frac{10267563344}{85865473} a^{18} - \frac{18681268559}{257596419} a^{17} - \frac{12403016392}{257596419} a^{16} + \frac{126129587750}{257596419} a^{15} - \frac{108706852533}{85865473} a^{14} + \frac{180482208595}{257596419} a^{13} + \frac{65016364200}{85865473} a^{12} + \frac{686360073866}{257596419} a^{11} - \frac{1628730592246}{257596419} a^{10} + \frac{1068399829907}{257596419} a^{9} - \frac{131443441969}{257596419} a^{8} + \frac{919263914954}{257596419} a^{7} - \frac{2564824514810}{257596419} a^{6} + \frac{880920894703}{85865473} a^{5} - \frac{1640266238186}{257596419} a^{4} + \frac{331721523585}{85865473} a^{3} - \frac{612024660836}{257596419} a^{2} + \frac{239459512025}{257596419} a - \frac{13024885462}{85865473} \),  \( \frac{69263414900}{1459713041} a^{19} - \frac{98990575245}{1459713041} a^{18} - \frac{58325263366}{1459713041} a^{17} - \frac{31661277263}{1459713041} a^{16} + \frac{399712883294}{1459713041} a^{15} - \frac{59794785242}{85865473} a^{14} + \frac{34595791711}{85865473} a^{13} + \frac{635014217009}{1459713041} a^{12} + \frac{2089633010243}{1459713041} a^{11} - \frac{5196500477196}{1459713041} a^{10} + \frac{3342715150435}{1459713041} a^{9} - \frac{345642874025}{1459713041} a^{8} + \frac{2808495426073}{1459713041} a^{7} - \frac{8099927885639}{1459713041} a^{6} + \frac{8320217357093}{1459713041} a^{5} - \frac{5054769095606}{1459713041} a^{4} + \frac{3072299576199}{1459713041} a^{3} - \frac{1912378527199}{1459713041} a^{2} + \frac{729363743977}{1459713041} a - \frac{110382905066}{1459713041} \),  \( \frac{274186238164}{4379139123} a^{19} - \frac{310114549069}{4379139123} a^{18} - \frac{289820724893}{4379139123} a^{17} - \frac{236659651055}{4379139123} a^{16} + \frac{488919187375}{1459713041} a^{15} - \frac{213678048997}{257596419} a^{14} + \frac{82738018919}{257596419} a^{13} + \frac{2550755438822}{4379139123} a^{12} + \frac{9055463962178}{4379139123} a^{11} - \frac{5850080813225}{1459713041} a^{10} + \frac{3075300799592}{1459713041} a^{9} - \frac{96825647013}{1459713041} a^{8} + \frac{3838817430477}{1459713041} a^{7} - \frac{28462645152622}{4379139123} a^{6} + \frac{25918129239158}{4379139123} a^{5} - \frac{15187044858281}{4379139123} a^{4} + \frac{9684955053521}{4379139123} a^{3} - \frac{5733087900083}{4379139123} a^{2} + \frac{1940777253562}{4379139123} a - \frac{253833225818}{4379139123} \),  \( \frac{256082710069}{4379139123} a^{19} - \frac{243715209961}{4379139123} a^{18} - \frac{302541568538}{4379139123} a^{17} - \frac{282098082329}{4379139123} a^{16} + \frac{434567912183}{1459713041} a^{15} - \frac{186927350236}{257596419} a^{14} + \frac{46814430611}{257596419} a^{13} + \frac{2395193759594}{4379139123} a^{12} + \frac{8877549763871}{4379139123} a^{11} - \frac{4900271844177}{1459713041} a^{10} + \frac{2140885146371}{1459713041} a^{9} + \frac{123470246761}{1459713041} a^{8} + \frac{3655388509456}{1459713041} a^{7} - \frac{24560345888668}{4379139123} a^{6} + \frac{20332936581794}{4379139123} a^{5} - \frac{11475972626885}{4379139123} a^{4} + \frac{7619314371542}{4379139123} a^{3} - \frac{4327311695096}{4379139123} a^{2} + \frac{1283844712198}{4379139123} a - \frac{132245407610}{4379139123} \),  \( \frac{622498112728}{4379139123} a^{19} - \frac{284365108256}{1459713041} a^{18} - \frac{530709946850}{4379139123} a^{17} - \frac{338622627874}{4379139123} a^{16} + \frac{3513221362835}{4379139123} a^{15} - \frac{176515499718}{85865473} a^{14} + \frac{290883354725}{257596419} a^{13} + \frac{1843851157256}{1459713041} a^{12} + \frac{19047338827496}{4379139123} a^{11} - \frac{45218319105220}{4379139123} a^{10} + \frac{29036826545753}{4379139123} a^{9} - \frac{3191303072413}{4379139123} a^{8} + \frac{25466279930543}{4379139123} a^{7} - \frac{71087980248518}{4379139123} a^{6} + \frac{24183797989670}{1459713041} a^{5} - \frac{44502536818610}{4379139123} a^{4} + \frac{9044874356995}{1459713041} a^{3} - \frac{16724636160620}{4379139123} a^{2} + \frac{6417441948464}{4379139123} a - \frac{336293108530}{1459713041} \),  \( \frac{12725910013}{85865473} a^{19} - \frac{51347100616}{257596419} a^{18} - \frac{11139794757}{85865473} a^{17} - \frac{22015499563}{257596419} a^{16} + \frac{214361547709}{257596419} a^{15} - \frac{547133431781}{257596419} a^{14} + \frac{97184943316}{85865473} a^{13} + \frac{341342269517}{257596419} a^{12} + \frac{392634562768}{85865473} a^{11} - \frac{2738969182184}{257596419} a^{10} + \frac{1726274751118}{257596419} a^{9} - \frac{176343190334}{257596419} a^{8} + \frac{1566953837377}{257596419} a^{7} - \frac{4317828222095}{257596419} a^{6} + \frac{4358348967530}{257596419} a^{5} - \frac{886503265379}{85865473} a^{4} + \frac{1628808176507}{257596419} a^{3} - \frac{333736532892}{85865473} a^{2} + \frac{379440591935}{257596419} a - \frac{58843172087}{257596419} \),  \( \frac{18442726496}{4379139123} a^{19} - \frac{28354740428}{1459713041} a^{18} + \frac{30422980973}{4379139123} a^{17} + \frac{69273759073}{4379139123} a^{16} + \frac{183771051676}{4379139123} a^{15} - \frac{10931905844}{85865473} a^{14} + \frac{49167233317}{257596419} a^{13} + \frac{34882336640}{1459713041} a^{12} - \frac{9595592843}{4379139123} a^{11} - \frac{3514809195605}{4379139123} a^{10} + \frac{3908271100021}{4379139123} a^{9} - \frac{1031329410416}{4379139123} a^{8} + \frac{472927679311}{4379139123} a^{7} - \frac{4719920728990}{4379139123} a^{6} + \frac{2473909747831}{1459713041} a^{5} - \frac{5100166634713}{4379139123} a^{4} + \frac{920844359084}{1459713041} a^{3} - \frac{1907247864193}{4379139123} a^{2} + \frac{944966254459}{4379139123} a - \frac{63404147661}{1459713041} \),  \( \frac{86929084073}{1459713041} a^{19} - \frac{365572249892}{4379139123} a^{18} - \frac{73619934518}{1459713041} a^{17} - \frac{130524656357}{4379139123} a^{16} + \frac{1489169676638}{4379139123} a^{15} - \frac{223555268912}{257596419} a^{14} + \frac{42160947849}{85865473} a^{13} + \frac{2355354812725}{4379139123} a^{12} + \frac{2642107206061}{1459713041} a^{11} - \frac{19268860492204}{4379139123} a^{10} + \frac{12389354382080}{4379139123} a^{9} - \frac{1350072886834}{4379139123} a^{8} + \frac{10617772774694}{4379139123} a^{7} - \frac{30166750940914}{4379139123} a^{6} + \frac{30895127472859}{4379139123} a^{5} - \frac{6294115831162}{1459713041} a^{4} + \frac{11501845134472}{4379139123} a^{3} - \frac{2373386563199}{1459713041} a^{2} + \frac{2734532475973}{4379139123} a - \frac{424558247398}{4379139123} \),  \( \frac{72693729430}{4379139123} a^{19} - \frac{6629273768}{4379139123} a^{18} - \frac{128570458199}{4379139123} a^{17} - \frac{55594378598}{1459713041} a^{16} + \frac{277424279866}{4379139123} a^{15} - \frac{35700099206}{257596419} a^{14} - \frac{27481249561}{257596419} a^{13} + \frac{684883114930}{4379139123} a^{12} + \frac{3117542290250}{4379139123} a^{11} - \frac{1832402393063}{4379139123} a^{10} - \frac{1140985615001}{4379139123} a^{9} + \frac{821328236032}{4379139123} a^{8} + \frac{3404469309307}{4379139123} a^{7} - \frac{1403976336323}{1459713041} a^{6} + \frac{511178507752}{4379139123} a^{5} + \frac{231512870782}{4379139123} a^{4} + \frac{350501921356}{4379139123} a^{3} + \frac{134265351574}{4379139123} a^{2} - \frac{102433391125}{1459713041} a + \frac{86468861213}{4379139123} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 99.9526828541 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_5\wr C_2$ (as 20T48):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.12176.1, 10.0.593019904.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
761Data not computed