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Properties

Label 20.4.16998350586...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{38}\cdot 5^{22}\cdot 11^{10}$
Root discriminant $72.70$
Ramified primes $2, 5, 11$
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![103710079859, 244797530, -681245695, 1120914990, -1217545665, 892866516, -473489850, 154609500, -43483725, 3220250, -644335, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 - 644335*x^10 + 3220250*x^9 - 43483725*x^8 + 154609500*x^7 - 473489850*x^6 + 892866516*x^5 - 1217545665*x^4 + 1120914990*x^3 - 681245695*x^2 + 244797530*x + 103710079859)
 
gp: K = bnfinit(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 - 644335*x^10 + 3220250*x^9 - 43483725*x^8 + 154609500*x^7 - 473489850*x^6 + 892866516*x^5 - 1217545665*x^4 + 1120914990*x^3 - 681245695*x^2 + 244797530*x + 103710079859, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 35 x^{18} - 30 x^{17} - 105 x^{16} + 228 x^{15} + 90 x^{14} - 540 x^{13} + 45 x^{12} + 770 x^{11} - 644335 x^{10} + 3220250 x^{9} - 43483725 x^{8} + 154609500 x^{7} - 473489850 x^{6} + 892866516 x^{5} - 1217545665 x^{4} + 1120914990 x^{3} - 681245695 x^{2} + 244797530 x + 103710079859 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(16998350586511360000000000000000000000=2^{38}\cdot 5^{22}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.70$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{11} a^{4} - \frac{2}{11} a^{3} - \frac{1}{11} a^{2} + \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{5} - \frac{5}{11} a^{3} + \frac{5}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{6} + \frac{1}{11} a^{3} + \frac{1}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{7} + \frac{2}{11} a^{3} + \frac{2}{11} a^{2} + \frac{3}{11} a - \frac{1}{11}$, $\frac{1}{121} a^{8} - \frac{4}{121} a^{7} + \frac{2}{121} a^{6} - \frac{3}{121} a^{5} - \frac{5}{121} a^{4} + \frac{47}{121} a^{3} + \frac{2}{121} a^{2} - \frac{51}{121} a - \frac{21}{121}$, $\frac{1}{121} a^{9} - \frac{3}{121} a^{7} + \frac{5}{121} a^{6} + \frac{5}{121} a^{5} + \frac{5}{121} a^{4} + \frac{25}{121} a^{3} + \frac{1}{121} a^{2} - \frac{5}{121} a + \frac{48}{121}$, $\frac{1}{605} a^{10} + \frac{3}{121} a^{7} - \frac{26}{605} a^{5} + \frac{2}{121} a^{4} + \frac{57}{121} a^{3} + \frac{9}{121} a^{2} - \frac{32}{121} a - \frac{63}{605}$, $\frac{1}{605} a^{11} + \frac{1}{121} a^{7} - \frac{1}{605} a^{6} - \frac{5}{121} a^{4} - \frac{5}{11} a^{3} + \frac{17}{121} a^{2} + \frac{152}{605} a + \frac{30}{121}$, $\frac{1}{6655} a^{12} + \frac{1}{1331} a^{11} - \frac{2}{6655} a^{10} + \frac{2}{1331} a^{9} + \frac{5}{1331} a^{8} + \frac{269}{6655} a^{7} + \frac{28}{1331} a^{6} + \frac{127}{6655} a^{5} - \frac{17}{1331} a^{4} + \frac{614}{1331} a^{3} - \frac{1498}{6655} a^{2} - \frac{67}{1331} a - \frac{2804}{6655}$, $\frac{1}{6655} a^{13} - \frac{1}{1331} a^{11} - \frac{2}{6655} a^{10} - \frac{5}{1331} a^{9} - \frac{21}{6655} a^{8} - \frac{32}{1331} a^{7} + \frac{57}{1331} a^{6} - \frac{258}{6655} a^{5} - \frac{16}{1331} a^{4} + \frac{2402}{6655} a^{3} - \frac{153}{1331} a^{2} + \frac{47}{1331} a + \frac{2811}{6655}$, $\frac{1}{6655} a^{14} + \frac{1}{6655} a^{11} - \frac{2}{6655} a^{10} - \frac{26}{6655} a^{9} + \frac{4}{1331} a^{8} + \frac{29}{1331} a^{7} + \frac{299}{6655} a^{6} - \frac{138}{6655} a^{5} - \frac{113}{6655} a^{4} + \frac{563}{1331} a^{3} - \frac{428}{1331} a^{2} + \frac{2687}{6655} a - \frac{2624}{6655}$, $\frac{1}{6655} a^{15} + \frac{4}{6655} a^{11} - \frac{2}{6655} a^{10} + \frac{2}{1331} a^{9} + \frac{2}{1331} a^{8} + \frac{50}{1331} a^{7} + \frac{96}{6655} a^{6} + \frac{123}{6655} a^{5} - \frac{47}{1331} a^{4} + \frac{630}{1331} a^{3} + \frac{331}{1331} a^{2} + \frac{2683}{6655} a - \frac{67}{6655}$, $\frac{1}{366025} a^{16} - \frac{8}{366025} a^{15} + \frac{9}{366025} a^{14} + \frac{2}{33275} a^{13} + \frac{18}{366025} a^{12} + \frac{12}{366025} a^{11} + \frac{156}{366025} a^{10} - \frac{714}{366025} a^{9} + \frac{183}{366025} a^{8} - \frac{10968}{366025} a^{7} - \frac{2011}{366025} a^{6} - \frac{7074}{366025} a^{5} + \frac{10083}{366025} a^{4} + \frac{3119}{33275} a^{3} - \frac{15589}{366025} a^{2} + \frac{114331}{366025} a - \frac{49444}{366025}$, $\frac{1}{366025} a^{17} - \frac{16}{366025} a^{14} - \frac{26}{366025} a^{13} - \frac{9}{366025} a^{12} + \frac{32}{366025} a^{11} + \frac{204}{366025} a^{10} - \frac{1294}{366025} a^{9} + \frac{131}{33275} a^{8} + \frac{947}{73205} a^{7} + \frac{6208}{366025} a^{6} - \frac{8724}{366025} a^{5} + \frac{6953}{366025} a^{4} + \frac{1043}{366025} a^{3} - \frac{51961}{366025} a^{2} - \frac{12486}{366025} a + \frac{118423}{366025}$, $\frac{1}{762818471882136266036310475} a^{18} - \frac{9}{762818471882136266036310475} a^{17} + \frac{81361735748538942544}{762818471882136266036310475} a^{16} - \frac{650893885988311540148}{762818471882136266036310475} a^{15} - \frac{53333823886437726082181}{762818471882136266036310475} a^{14} + \frac{40857325214155733158408}{762818471882136266036310475} a^{13} - \frac{2806142047324666456129}{152563694376427253207262095} a^{12} + \frac{555033616810358691656689}{762818471882136266036310475} a^{11} - \frac{21757673849880023468956}{762818471882136266036310475} a^{10} - \frac{2609520020011037885458984}{762818471882136266036310475} a^{9} + \frac{2860760581375450671318853}{762818471882136266036310475} a^{8} + \frac{18990367477291249497309601}{762818471882136266036310475} a^{7} + \frac{1002957383711157031535392}{30512738875285450641452419} a^{6} - \frac{27018704448855601568235327}{762818471882136266036310475} a^{5} - \frac{1492380226576055178795272}{762818471882136266036310475} a^{4} - \frac{344522124999118129484000372}{762818471882136266036310475} a^{3} - \frac{192685304071353579558093678}{762818471882136266036310475} a^{2} - \frac{43050167480386790097540724}{762818471882136266036310475} a - \frac{203375046799796715867701408}{762818471882136266036310475}$, $\frac{1}{251599900152779540420504177387058425} a^{19} + \frac{164914652}{251599900152779540420504177387058425} a^{18} + \frac{15817454779052710567669642269}{22872718195707230947318561580641675} a^{17} - \frac{90745341348551838182014742944}{251599900152779540420504177387058425} a^{16} - \frac{9905965981707638483468417054779}{251599900152779540420504177387058425} a^{15} - \frac{13646570384252026889027639645392}{251599900152779540420504177387058425} a^{14} - \frac{3641853871697058520754976102641}{251599900152779540420504177387058425} a^{13} - \frac{11425309252363371573037727545097}{251599900152779540420504177387058425} a^{12} + \frac{12218876475330135367833996101076}{22872718195707230947318561580641675} a^{11} - \frac{180885098476655837419008279948439}{251599900152779540420504177387058425} a^{10} - \frac{422171422658863748567040733003202}{251599900152779540420504177387058425} a^{9} + \frac{361330657501502633637589332663353}{251599900152779540420504177387058425} a^{8} - \frac{6568478722571799646363290750331834}{251599900152779540420504177387058425} a^{7} - \frac{1774901550409148219477391284091872}{50319980030555908084100835477411685} a^{6} + \frac{264615581531421398884707874544864}{50319980030555908084100835477411685} a^{5} - \frac{213669992298903922777298740970767}{251599900152779540420504177387058425} a^{4} - \frac{51786067735988736773486998139390658}{251599900152779540420504177387058425} a^{3} + \frac{85494293168239390007681795758779559}{251599900152779540420504177387058425} a^{2} + \frac{75578182842356989438966934825419244}{251599900152779540420504177387058425} a - \frac{93890244680629565757261816920769321}{251599900152779540420504177387058425}$

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:  $11$
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

\(\Q(\sqrt{22}) \), \(\Q(\sqrt{110}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{22})\), 5.1.50000.1, 10.2.824581120000000000.2, 10.2.4122905600000000000.1, 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
5Data not computed
$11$11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$