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Properties

Label 20.0.33767941114...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{22}\cdot 43^{10}$
Root discriminant $67.05$
Ramified primes $2, 5, 43$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10110844204, -183407180, 11358886060, -84886680, 5741435880, -3637742, 1673896860, 2386790, 309021980, 362580, 37717545, 17730, 3087345, 190, 167660, -4, 5790, 0, 115, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 115*x^18 + 5790*x^16 - 4*x^15 + 167660*x^14 + 190*x^13 + 3087345*x^12 + 17730*x^11 + 37717545*x^10 + 362580*x^9 + 309021980*x^8 + 2386790*x^7 + 1673896860*x^6 - 3637742*x^5 + 5741435880*x^4 - 84886680*x^3 + 11358886060*x^2 - 183407180*x + 10110844204)
 
gp: K = bnfinit(x^20 + 115*x^18 + 5790*x^16 - 4*x^15 + 167660*x^14 + 190*x^13 + 3087345*x^12 + 17730*x^11 + 37717545*x^10 + 362580*x^9 + 309021980*x^8 + 2386790*x^7 + 1673896860*x^6 - 3637742*x^5 + 5741435880*x^4 - 84886680*x^3 + 11358886060*x^2 - 183407180*x + 10110844204, 1)
 

Normalized defining polynomial

\( x^{20} + 115 x^{18} + 5790 x^{16} - 4 x^{15} + 167660 x^{14} + 190 x^{13} + 3087345 x^{12} + 17730 x^{11} + 37717545 x^{10} + 362580 x^{9} + 309021980 x^{8} + 2386790 x^{7} + 1673896860 x^{6} - 3637742 x^{5} + 5741435880 x^{4} - 84886680 x^{3} + 11358886060 x^{2} - 183407180 x + 10110844204 \)

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:  \(3376794111450663906250000000000000000=2^{16}\cdot 5^{22}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.05$
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:  $2, 5, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} - \frac{1}{14} a^{16} + \frac{3}{14} a^{13} + \frac{1}{14} a^{12} + \frac{1}{14} a^{11} - \frac{1}{7} a^{8} - \frac{1}{2} a^{7} - \frac{1}{14} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{72031836290521690918178863124974877467827995159965212141158112133777098} a^{19} - \frac{1916221328387848613446265990986741252015717526659295595165752511758261}{72031836290521690918178863124974877467827995159965212141158112133777098} a^{18} - \frac{570239820296128330975015859393144964954668298942051507158792949609617}{2483856423811092790282029762930168188545792936550524556591659039095762} a^{17} + \frac{265486272608834692507955301434518085838117197975753040512301312145436}{1241928211905546395141014881465084094272896468275262278295829519547881} a^{16} + \frac{1654627518103864517401934449631936442655681377278573843363166054304931}{10290262327217384416882694732139268209689713594280744591594016019111014} a^{15} - \frac{155134391651195331981223718636624760965317631395591731071674144294120}{1895574639224255024162601661183549407048105135788558214241002950888871} a^{14} - \frac{1154790046540063837282640168162097170134318549087473670790621844529468}{12005306048420281819696477187495812911304665859994202023526352022296183} a^{13} + \frac{1118564363763260693683669949751512739286507650744127351038377861910245}{5145131163608692208441347366069634104844856797140372295797008009555507} a^{12} + \frac{9103124712472916808834069750446499641265842432145543798479730785959585}{72031836290521690918178863124974877467827995159965212141158112133777098} a^{11} + \frac{17949982752579764376915491913798395427085857507054779467141980369020}{270796377032036432023228808740507058149729305112651173463000421555553} a^{10} + \frac{1448590848251886103275367714449268761096683223707886267614343699904175}{36015918145260845459089431562487438733913997579982606070579056066888549} a^{9} + \frac{2555011011099791688987701387810416991642826303034200892446472326450941}{36015918145260845459089431562487438733913997579982606070579056066888549} a^{8} - \frac{161527861769953121480378400859831714958282566148125674882291085357629}{827952141270364263427343254310056062848597645516841518863886346365254} a^{7} + \frac{6196158769455309849903970826325746981499534611622393067181267053877153}{72031836290521690918178863124974877467827995159965212141158112133777098} a^{6} - \frac{1656339949468034282899274511314434536120831829842476976428815418055895}{10290262327217384416882694732139268209689713594280744591594016019111014} a^{5} + \frac{4165736843114575877455375142052819347922402016885884948710062462944423}{12005306048420281819696477187495812911304665859994202023526352022296183} a^{4} + \frac{3288830265793232375865818985075717055751543447286986903380433593651717}{12005306048420281819696477187495812911304665859994202023526352022296183} a^{3} + \frac{1830595672276309774262327608700392349598449691695836727438701390907166}{12005306048420281819696477187495812911304665859994202023526352022296183} a^{2} + \frac{2073329434989448797663572239878915051790841952765011705396860201556540}{5145131163608692208441347366069634104844856797140372295797008009555507} a - \frac{7849572240247000505828171856708877677795416610310200404550927034724414}{36015918145260845459089431562487438733913997579982606070579056066888549}$

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 20T13):

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

\(\Q(\sqrt{-215}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-43})\), 5.1.50000.1, 10.0.1837605537500000000.1, 10.0.367521107500000000.1, 10.2.12500000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$