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Properties

Label 20.0.78130637554...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{9}\cdot 199^{5}\cdot 401^{5}$
Root discriminant $221.13$
Ramified primes $2, 5, 11, 199, 401$
Class number $32$ (GRH)
Class group $[2, 2, 2, 4]$ (GRH)
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![676348299686920069, 0, 184040238341354934, 0, 15138792734705123, 0, 361572835120248, 0, 3672413853363, 0, 87853088411, 0, 2068594443, 0, 22839038, 0, 138365, 0, 529, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 529*x^18 + 138365*x^16 + 22839038*x^14 + 2068594443*x^12 + 87853088411*x^10 + 3672413853363*x^8 + 361572835120248*x^6 + 15138792734705123*x^4 + 184040238341354934*x^2 + 676348299686920069)
 
gp: K = bnfinit(x^20 + 529*x^18 + 138365*x^16 + 22839038*x^14 + 2068594443*x^12 + 87853088411*x^10 + 3672413853363*x^8 + 361572835120248*x^6 + 15138792734705123*x^4 + 184040238341354934*x^2 + 676348299686920069, 1)
 

Normalized defining polynomial

\( x^{20} + 529 x^{18} + 138365 x^{16} + 22839038 x^{14} + 2068594443 x^{12} + 87853088411 x^{10} + 3672413853363 x^{8} + 361572835120248 x^{6} + 15138792734705123 x^{4} + 184040238341354934 x^{2} + 676348299686920069 \)

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:  \(78130637554327275128616443231859804160000000000=2^{20}\cdot 5^{10}\cdot 11^{9}\cdot 199^{5}\cdot 401^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $221.13$
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:  $2, 5, 11, 199, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{2}{11} a^{12} + \frac{2}{11} a^{10} - \frac{4}{11} a^{8} + \frac{4}{11} a^{6} - \frac{1}{11} a^{4} + \frac{5}{11} a^{2}$, $\frac{1}{11} a^{15} - \frac{2}{11} a^{13} + \frac{2}{11} a^{11} - \frac{4}{11} a^{9} + \frac{4}{11} a^{7} - \frac{1}{11} a^{5} + \frac{5}{11} a^{3}$, $\frac{1}{877789} a^{16} + \frac{529}{877789} a^{14} + \frac{138365}{877789} a^{12} + \frac{16524}{877789} a^{10} - \frac{354230}{877789} a^{8} - \frac{38514}{79799} a^{6} + \frac{113962}{877789} a^{4} - \frac{416258}{877789} a^{2}$, $\frac{1}{877789} a^{17} + \frac{529}{877789} a^{15} + \frac{138365}{877789} a^{13} + \frac{16524}{877789} a^{11} - \frac{354230}{877789} a^{9} - \frac{38514}{79799} a^{7} + \frac{113962}{877789} a^{5} - \frac{416258}{877789} a^{3}$, $\frac{1}{1506180233621636210228899378660058724082461942797175301220971467749577879463306} a^{18} + \frac{43152874011222967564005842076517162930959824656199612058948967096843870}{753090116810818105114449689330029362041230971398587650610485733874788939731653} a^{16} + \frac{66420163783991973397196909689879979922341798692167179639695083467362230212985}{1506180233621636210228899378660058724082461942797175301220971467749577879463306} a^{14} - \frac{76904897996917911202970595148344133902995413975505530017959265071917024310667}{1506180233621636210228899378660058724082461942797175301220971467749577879463306} a^{12} - \frac{147501155187651937227339360654570596554997307438197638368587036510873586716692}{753090116810818105114449689330029362041230971398587650610485733874788939731653} a^{10} + \frac{31229186219414125270593917096320386000666334960344912602257197736701807277937}{136925475783785110020809034423641702189314722072470481929179224340870716314846} a^{8} + \frac{163418688471727283126866946397770785836971777363824120795175297936944244817988}{753090116810818105114449689330029362041230971398587650610485733874788939731653} a^{6} - \frac{348606739012146794357637341439100374155480855795201137885262468125450520860799}{753090116810818105114449689330029362041230971398587650610485733874788939731653} a^{4} - \frac{363093655771021809290609914464976736989473518043133109013700944501436209}{1715879594779196606734533445577534833635944336050207169628431739005134354} a^{2} - \frac{572545718885743803808729950317925297422910450042677972281911649737}{1954774546934623932100463147268346759455796707466381066097241750586}$, $\frac{1}{1506180233621636210228899378660058724082461942797175301220971467749577879463306} a^{19} + \frac{43152874011222967564005842076517162930959824656199612058948967096843870}{753090116810818105114449689330029362041230971398587650610485733874788939731653} a^{17} + \frac{66420163783991973397196909689879979922341798692167179639695083467362230212985}{1506180233621636210228899378660058724082461942797175301220971467749577879463306} a^{15} - \frac{76904897996917911202970595148344133902995413975505530017959265071917024310667}{1506180233621636210228899378660058724082461942797175301220971467749577879463306} a^{13} - \frac{147501155187651937227339360654570596554997307438197638368587036510873586716692}{753090116810818105114449689330029362041230971398587650610485733874788939731653} a^{11} + \frac{31229186219414125270593917096320386000666334960344912602257197736701807277937}{136925475783785110020809034423641702189314722072470481929179224340870716314846} a^{9} + \frac{163418688471727283126866946397770785836971777363824120795175297936944244817988}{753090116810818105114449689330029362041230971398587650610485733874788939731653} a^{7} - \frac{348606739012146794357637341439100374155480855795201137885262468125450520860799}{753090116810818105114449689330029362041230971398587650610485733874788939731653} a^{5} - \frac{363093655771021809290609914464976736989473518043133109013700944501436209}{1715879594779196606734533445577534833635944336050207169628431739005134354} a^{3} - \frac{572545718885743803808729950317925297422910450042677972281911649737}{1954774546934623932100463147268346759455796707466381066097241750586} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:  $9$
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:  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:  \( 36289305879900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.331913965625.1

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 32 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.6.3.2$x^{6} - 39601 x^{2} + 31522396$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
199.8.0.1$x^{8} - x + 15$$1$$8$$0$$C_8$$[\ ]^{8}$
401Data not computed