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Properties

Label 20.4.11145982994...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{22}\cdot 61^{10}$
Root discriminant $79.86$
Ramified primes $2, 5, 61$
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![1096989910724, 5476545300, -624326270020, -824749960, 162923648340, -11303998, -25731161080, 10059910, 2725736800, -849540, -202311647, 30210, 10648925, -330, -392380, -4, 9690, 0, -145, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 145*x^18 + 9690*x^16 - 4*x^15 - 392380*x^14 - 330*x^13 + 10648925*x^12 + 30210*x^11 - 202311647*x^10 - 849540*x^9 + 2725736800*x^8 + 10059910*x^7 - 25731161080*x^6 - 11303998*x^5 + 162923648340*x^4 - 824749960*x^3 - 624326270020*x^2 + 5476545300*x + 1096989910724)
 
gp: K = bnfinit(x^20 - 145*x^18 + 9690*x^16 - 4*x^15 - 392380*x^14 - 330*x^13 + 10648925*x^12 + 30210*x^11 - 202311647*x^10 - 849540*x^9 + 2725736800*x^8 + 10059910*x^7 - 25731161080*x^6 - 11303998*x^5 + 162923648340*x^4 - 824749960*x^3 - 624326270020*x^2 + 5476545300*x + 1096989910724, 1)
 

Normalized defining polynomial

\( x^{20} - 145 x^{18} + 9690 x^{16} - 4 x^{15} - 392380 x^{14} - 330 x^{13} + 10648925 x^{12} + 30210 x^{11} - 202311647 x^{10} - 849540 x^{9} + 2725736800 x^{8} + 10059910 x^{7} - 25731161080 x^{6} - 11303998 x^{5} + 162923648340 x^{4} - 824749960 x^{3} - 624326270020 x^{2} + 5476545300 x + 1096989910724 \)

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:  \(111459829947325406406250000000000000000=2^{16}\cdot 5^{22}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.86$
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:  $2, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} + \frac{3}{14} a^{16} + \frac{1}{14} a^{15} + \frac{3}{14} a^{14} + \frac{1}{7} a^{12} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{5}{14} a^{8} + \frac{3}{14} a^{7} - \frac{2}{7} a^{6} - \frac{5}{14} a^{5} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{9926255053023636160768374353173761916416220772508702145915606885081514043478886} a^{19} + \frac{64765134502868890249101786832026968150653800254449346442529984205979000000714}{4963127526511818080384187176586880958208110386254351072957803442540757021739443} a^{18} - \frac{613840169571789896872574850670011250988697198243852088660089813606329368560463}{4963127526511818080384187176586880958208110386254351072957803442540757021739443} a^{17} - \frac{1111972176179963818282411609640221656465368509712659981476550990756945792112235}{9926255053023636160768374353173761916416220772508702145915606885081514043478886} a^{16} + \frac{793293550985912485046869488126328753338291619288888308444871841943348503657645}{9926255053023636160768374353173761916416220772508702145915606885081514043478886} a^{15} + \frac{206770036490621254121550771813216775348809671522065697223146267123675062229033}{1418036436146233737252624907596251702345174396072671735130800983583073434782698} a^{14} - \frac{1147642790103503035319481409156560741928869305253775187607501897671574486787891}{9926255053023636160768374353173761916416220772508702145915606885081514043478886} a^{13} - \frac{89909449929484166161454224442468035910143901908837389786759149469455418221404}{709018218073116868626312453798125851172587198036335867565400491791536717391349} a^{12} - \frac{681968944003452335227354360105626229083564545679098410776002293646239308219487}{9926255053023636160768374353173761916416220772508702145915606885081514043478886} a^{11} + \frac{1677013966039239295174707154173228152151795056127578434889991962284656089548523}{9926255053023636160768374353173761916416220772508702145915606885081514043478886} a^{10} - \frac{3083974967426824278578615519002838620184308405267615728531843324999933228801333}{9926255053023636160768374353173761916416220772508702145915606885081514043478886} a^{9} - \frac{542670352994949661062583065810721701736460750610217731062515264129914314251615}{9926255053023636160768374353173761916416220772508702145915606885081514043478886} a^{8} - \frac{2805009825104961407497859142570887868760376661552568706783273124886616069644671}{9926255053023636160768374353173761916416220772508702145915606885081514043478886} a^{7} - \frac{477793679195705263074920744108224720140211292417958126334918407830625288627804}{4963127526511818080384187176586880958208110386254351072957803442540757021739443} a^{6} + \frac{1661344052897900906173222445293519412680415059630209723219818536005925296958575}{4963127526511818080384187176586880958208110386254351072957803442540757021739443} a^{5} - \frac{378110386064300340203644568413193870879860272243328346425353673757998107323544}{4963127526511818080384187176586880958208110386254351072957803442540757021739443} a^{4} + \frac{641220426823203297671328772087172081099933738212770503883354709103295765728996}{4963127526511818080384187176586880958208110386254351072957803442540757021739443} a^{3} - \frac{412230789354122890655835347420947710640397280269895472832875614443751276305610}{4963127526511818080384187176586880958208110386254351072957803442540757021739443} a^{2} - \frac{2467003002355786512243203772145588075458553653938509992025616017423778440946640}{4963127526511818080384187176586880958208110386254351072957803442540757021739443} a + \frac{242802758672197674019705397885111057520053232899946295280742861175739035847877}{709018218073116868626312453798125851172587198036335867565400491791536717391349}$

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

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{305}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{61})\), 5.1.50000.1, 10.2.10557453762500000000.1, 10.2.2111490752500000000.1, 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.8.1$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}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$61$61.10.5.1$x^{10} - 7442 x^{6} + 13845841 x^{2} - 30405466836$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
61.10.5.1$x^{10} - 7442 x^{6} + 13845841 x^{2} - 30405466836$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$