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Properties

Label 20.0.71319704189...8125.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{31}\cdot 11^{10}$
Root discriminant $69.61$
Ramified primes $3, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![128412299536, 174097207960, 125265538040, 55400347980, 13088961695, -1385834729, -2438250905, -1145438325, -259823140, -24508695, 13220076, 5100880, 1449605, 128940, 24430, -2939, 1090, -95, 45, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 45*x^18 - 95*x^17 + 1090*x^16 - 2939*x^15 + 24430*x^14 + 128940*x^13 + 1449605*x^12 + 5100880*x^11 + 13220076*x^10 - 24508695*x^9 - 259823140*x^8 - 1145438325*x^7 - 2438250905*x^6 - 1385834729*x^5 + 13088961695*x^4 + 55400347980*x^3 + 125265538040*x^2 + 174097207960*x + 128412299536)
 
gp: K = bnfinit(x^20 - 5*x^19 + 45*x^18 - 95*x^17 + 1090*x^16 - 2939*x^15 + 24430*x^14 + 128940*x^13 + 1449605*x^12 + 5100880*x^11 + 13220076*x^10 - 24508695*x^9 - 259823140*x^8 - 1145438325*x^7 - 2438250905*x^6 - 1385834729*x^5 + 13088961695*x^4 + 55400347980*x^3 + 125265538040*x^2 + 174097207960*x + 128412299536, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 45 x^{18} - 95 x^{17} + 1090 x^{16} - 2939 x^{15} + 24430 x^{14} + 128940 x^{13} + 1449605 x^{12} + 5100880 x^{11} + 13220076 x^{10} - 24508695 x^{9} - 259823140 x^{8} - 1145438325 x^{7} - 2438250905 x^{6} - 1385834729 x^{5} + 13088961695 x^{4} + 55400347980 x^{3} + 125265538040 x^{2} + 174097207960 x + 128412299536 \)

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:  \(7131970418917243368923664093017578125=3^{10}\cdot 5^{31}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.61$
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:  $3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{12} a^{6} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{5}{12} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{15} + \frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{6} a^{8} + \frac{1}{12} a^{7} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{5}{12} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{36} a^{16} + \frac{1}{36} a^{14} - \frac{1}{18} a^{13} - \frac{1}{12} a^{12} - \frac{1}{6} a^{11} - \frac{1}{18} a^{10} + \frac{1}{9} a^{9} + \frac{2}{9} a^{8} + \frac{1}{9} a^{7} + \frac{13}{36} a^{6} - \frac{1}{18} a^{5} - \frac{17}{36} a^{4} + \frac{1}{6} a^{3} - \frac{1}{36} a^{2} - \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{36} a^{17} + \frac{1}{36} a^{15} + \frac{1}{36} a^{14} - \frac{1}{12} a^{13} - \frac{2}{9} a^{11} + \frac{7}{36} a^{10} - \frac{1}{9} a^{9} + \frac{1}{36} a^{8} + \frac{1}{36} a^{7} - \frac{17}{36} a^{6} + \frac{13}{36} a^{5} + \frac{1}{3} a^{4} - \frac{1}{36} a^{3} - \frac{13}{36} a^{2} - \frac{1}{18} a$, $\frac{1}{468} a^{18} + \frac{1}{468} a^{16} + \frac{1}{468} a^{15} - \frac{5}{156} a^{14} + \frac{1}{13} a^{13} - \frac{1}{234} a^{12} + \frac{37}{468} a^{11} + \frac{23}{117} a^{10} + \frac{7}{468} a^{9} - \frac{23}{468} a^{8} + \frac{55}{468} a^{7} + \frac{73}{468} a^{6} - \frac{7}{39} a^{5} - \frac{199}{468} a^{4} - \frac{61}{468} a^{3} + \frac{37}{117} a^{2} - \frac{29}{78} a - \frac{8}{39}$, $\frac{1}{22627806069279747626302913070480330687473749085948915811984478589356109065467891349645623336} a^{19} - \frac{10841087859268859216404520150338509184648169131613487876364308000474763409855090237475151}{22627806069279747626302913070480330687473749085948915811984478589356109065467891349645623336} a^{18} - \frac{190054180855146361897848124677653964696199275090399570039773514964526313018358561628628661}{22627806069279747626302913070480330687473749085948915811984478589356109065467891349645623336} a^{17} + \frac{210397440343076131197295356749334592771758514312702641635935719722776430570719718175178013}{22627806069279747626302913070480330687473749085948915811984478589356109065467891349645623336} a^{16} - \frac{4934639417208409378562792576968788132208613265434784376442915891697589787594457641150957}{290100077811278815721832218852311931890689090845498920666467674222514218788049889098020812} a^{15} + \frac{182092187521848930169081394472966257761411883312547683249594270051715820952790334871045801}{7542602023093249208767637690160110229157916361982971937328159529785369688489297116548541112} a^{14} - \frac{31240965134950442802541017050138658458762306933581865711326025733710095492966448212203656}{942825252886656151095954711270013778644739545247871492166019941223171211061162139568567639} a^{13} - \frac{222146663842678214035105893662166016200589320389190167882173421383729899285997954609764835}{3771301011546624604383818845080055114578958180991485968664079764892684844244648558274270556} a^{12} - \frac{5624982608460936879528430149359885438538443059930189053077889194874728065470168038859267253}{22627806069279747626302913070480330687473749085948915811984478589356109065467891349645623336} a^{11} - \frac{2312412373621037679509128327368509405262896703494146951427781168495117060743851253063133233}{11313903034639873813151456535240165343736874542974457905992239294678054532733945674822811668} a^{10} - \frac{2112573518215862561379934720917818536143535938425707644282356919858381676365756917869451329}{11313903034639873813151456535240165343736874542974457905992239294678054532733945674822811668} a^{9} - \frac{1810754339188928041892542589370941995657345342162283144200655356765704779776092449263569247}{22627806069279747626302913070480330687473749085948915811984478589356109065467891349645623336} a^{8} - \frac{18072849564571757995792927562170286653852350064357987038522734540682876581958072590294739}{314275084295552050365318237090004592881579848415957164055339980407723737020387379856189213} a^{7} + \frac{6038258406631185109570332265417379832451844762148621240197379372852097203617391939892389401}{22627806069279747626302913070480330687473749085948915811984478589356109065467891349645623336} a^{6} - \frac{10256911437474503015411671354180061234508763103573081945980472349538746744083325338885649319}{22627806069279747626302913070480330687473749085948915811984478589356109065467891349645623336} a^{5} + \frac{2400329539641970256348988126561067790886797459637818755140856652547852705886289208876742919}{22627806069279747626302913070480330687473749085948915811984478589356109065467891349645623336} a^{4} - \frac{560627517873919034142875465756011812817776177732910900575447346991470053965821387137366437}{1740600466867672894330993313113871591344134545072993523998806045335085312728299334588124872} a^{3} + \frac{1180162766974325871600588747322225008124695850751131652322097010879373150837308224651645822}{2828475758659968453287864133810041335934218635743614476498059823669513633183486418705702917} a^{2} - \frac{47912026203724000236284363286634589807255839901595326430549490722849494097892723566285224}{148867145192629918594098112305791649259695717670716551394634727561553349114920337826615943} a - \frac{772217577701353642828392941781146463269425714707798520223361243038842756579377432209663723}{2828475758659968453287864133810041335934218635743614476498059823669513633183486418705702917}$

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:  $9$
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

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.136125.2, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$11$11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$