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Properties

Label 20.0.40431106071...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{26}\cdot 11^{16}$
Root discriminant $95.57$
Ramified primes $3, 5, 11$
Class number $134480$ (GRH)
Class group $[2, 2, 82, 410]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![960400, 0, 7570500, 0, 55495925, 0, 31575125, 0, 12738625, 0, 2291230, 0, 290350, 0, 22970, 0, 1325, 0, 45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 45*x^18 + 1325*x^16 + 22970*x^14 + 290350*x^12 + 2291230*x^10 + 12738625*x^8 + 31575125*x^6 + 55495925*x^4 + 7570500*x^2 + 960400)
 
gp: K = bnfinit(x^20 + 45*x^18 + 1325*x^16 + 22970*x^14 + 290350*x^12 + 2291230*x^10 + 12738625*x^8 + 31575125*x^6 + 55495925*x^4 + 7570500*x^2 + 960400, 1)
 

Normalized defining polynomial

\( x^{20} + 45 x^{18} + 1325 x^{16} + 22970 x^{14} + 290350 x^{12} + 2291230 x^{10} + 12738625 x^{8} + 31575125 x^{6} + 55495925 x^{4} + 7570500 x^{2} + 960400 \)

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:  \(4043110607138384185566008090972900390625=3^{10}\cdot 5^{26}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.57$
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 Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{3}$, $\frac{1}{40} a^{10} - \frac{1}{8} a^{4} - \frac{1}{2} a$, $\frac{1}{40} a^{11} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{240} a^{12} + \frac{1}{120} a^{10} - \frac{1}{16} a^{9} + \frac{11}{48} a^{6} + \frac{1}{8} a^{4} + \frac{1}{16} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6}$, $\frac{1}{1680} a^{13} + \frac{1}{840} a^{11} - \frac{1}{80} a^{10} + \frac{3}{56} a^{9} + \frac{83}{336} a^{7} + \frac{5}{56} a^{5} - \frac{3}{16} a^{4} + \frac{23}{168} a^{3} - \frac{1}{2} a^{2} - \frac{4}{21} a$, $\frac{1}{5040} a^{14} + \frac{1}{560} a^{12} - \frac{1}{80} a^{11} + \frac{31}{2520} a^{10} - \frac{1}{16} a^{9} + \frac{41}{1008} a^{8} - \frac{1}{4} a^{7} + \frac{107}{1008} a^{6} - \frac{3}{16} a^{5} + \frac{65}{504} a^{4} + \frac{1}{16} a^{3} - \frac{41}{168} a^{2} - \frac{1}{4} a - \frac{1}{9}$, $\frac{1}{5040} a^{15} + \frac{11}{1260} a^{11} - \frac{29}{504} a^{9} - \frac{17}{126} a^{7} - \frac{5}{36} a^{5} - \frac{73}{336} a^{3} - \frac{5}{126} a - \frac{1}{2}$, $\frac{1}{96647040} a^{16} - \frac{1193}{24161760} a^{14} + \frac{2371}{3020220} a^{12} - \frac{1}{80} a^{11} - \frac{82931}{16107840} a^{10} - \frac{1}{16} a^{9} - \frac{56069}{1208088} a^{8} + \frac{80839}{402696} a^{6} - \frac{3}{16} a^{5} - \frac{1324163}{19329408} a^{4} - \frac{7}{16} a^{3} + \frac{229919}{690336} a^{2} - \frac{1}{2} a - \frac{30601}{172584}$, $\frac{1}{96647040} a^{17} - \frac{1193}{24161760} a^{15} + \frac{2293}{12080880} a^{13} - \frac{102107}{16107840} a^{11} - \frac{90565}{2416176} a^{9} - \frac{37273}{805392} a^{7} - \frac{3050003}{19329408} a^{5} - \frac{1770337}{4832352} a^{3} - \frac{588139}{1208088} a - \frac{1}{2}$, $\frac{1}{163502697121972911360} a^{18} - \frac{1}{193294080} a^{17} + \frac{459223420883}{163502697121972911360} a^{16} - \frac{3601}{48323520} a^{15} + \frac{16922137136369}{267161269807145280} a^{14} - \frac{2293}{24161760} a^{13} + \frac{7055542792327637}{3554406459173324160} a^{12} + \frac{121393}{10738560} a^{11} + \frac{359102847075249853}{81751348560986455680} a^{10} + \frac{229591}{4832352} a^{9} - \frac{48488543349228091}{1021891857012330696} a^{8} + \frac{145937}{1610784} a^{7} - \frac{227743933455498715}{1923561142611446016} a^{6} - \frac{6346237}{38658816} a^{5} - \frac{1211629410179067683}{10900179808131527424} a^{4} + \frac{402889}{1380672} a^{3} - \frac{3338599163802616205}{8175134856098645568} a^{2} + \frac{636079}{2416176} a - \frac{10995177937747117}{41709871714789008}$, $\frac{1}{1144518879853810379520} a^{19} + \frac{43503292285}{38150629328460345984} a^{17} - \frac{1}{193294080} a^{16} - \frac{2646355389512267}{71532429990863148720} a^{15} - \frac{3601}{48323520} a^{14} - \frac{62941726572919}{1463579130247839360} a^{13} - \frac{31057}{24161760} a^{12} + \frac{852911762860961777}{286129719963452594880} a^{11} + \frac{19165}{2147712} a^{10} + \frac{739165603503809683}{28612971996345259488} a^{9} - \frac{288161}{4832352} a^{8} - \frac{30321607329059717267}{228903775970762075904} a^{7} - \frac{247171}{1610784} a^{6} + \frac{3656545573807819439}{114451887985381037952} a^{5} + \frac{6079811}{38658816} a^{4} - \frac{256544302062598135}{9537657332115086496} a^{3} - \frac{4658417}{9664704} a^{2} - \frac{45580540291760473}{145984551001761528} a + \frac{136069}{345168}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{82}\times C_{410}$, which has order $134480$ (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:  \( -\frac{13088843577}{89053756602381760} a^{18} - \frac{351862678127}{53432253961429056} a^{16} - \frac{2584981588705}{13358063490357264} a^{14} - \frac{3880398475531}{1161570738291936} a^{12} - \frac{5622618918697657}{133580634903572640} a^{10} - \frac{91743977896603}{278292989382443} a^{8} - \frac{97346641348624295}{53432253961429056} a^{6} - \frac{234272859580158983}{53432253961429056} a^{4} - \frac{105522713889382655}{13358063490357264} a^{2} - \frac{5288690925091}{68153385154884} \) (order $6$)
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:  \( 143712739.876 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 5.5.228765625.1 x5, 10.10.261668555908203125.1, 10.0.63585459085693359375.3 x5, 10.0.12717091817138671875.1 x5

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{10}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5Data not computed
$11$11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$