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Properties

Label 20.0.14227232712...2976.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 19^{10}\cdot 43^{10}$
Root discriminant $80.85$
Ramified primes $2, 19, 43$
Class number $18750$ (GRH)
Class group $[5, 5, 5, 150]$ (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![53218751, -105586732, 201449168, -203873920, 180714223, -119636172, 70051658, -34282576, 15156257, -5913476, 2100216, -689048, 215150, -61676, 16802, -3852, 939, -168, 34, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 34*x^18 - 168*x^17 + 939*x^16 - 3852*x^15 + 16802*x^14 - 61676*x^13 + 215150*x^12 - 689048*x^11 + 2100216*x^10 - 5913476*x^9 + 15156257*x^8 - 34282576*x^7 + 70051658*x^6 - 119636172*x^5 + 180714223*x^4 - 203873920*x^3 + 201449168*x^2 - 105586732*x + 53218751)
 
gp: K = bnfinit(x^20 - 4*x^19 + 34*x^18 - 168*x^17 + 939*x^16 - 3852*x^15 + 16802*x^14 - 61676*x^13 + 215150*x^12 - 689048*x^11 + 2100216*x^10 - 5913476*x^9 + 15156257*x^8 - 34282576*x^7 + 70051658*x^6 - 119636172*x^5 + 180714223*x^4 - 203873920*x^3 + 201449168*x^2 - 105586732*x + 53218751, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 34 x^{18} - 168 x^{17} + 939 x^{16} - 3852 x^{15} + 16802 x^{14} - 61676 x^{13} + 215150 x^{12} - 689048 x^{11} + 2100216 x^{10} - 5913476 x^{9} + 15156257 x^{8} - 34282576 x^{7} + 70051658 x^{6} - 119636172 x^{5} + 180714223 x^{4} - 203873920 x^{3} + 201449168 x^{2} - 105586732 x + 53218751 \)

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:  \(142272327122261506049033808572607102976=2^{30}\cdot 19^{10}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.85$
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:  $2, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{190} a^{16} - \frac{3}{38} a^{15} + \frac{1}{5} a^{14} + \frac{39}{190} a^{13} + \frac{3}{38} a^{12} + \frac{12}{95} a^{11} + \frac{1}{95} a^{10} - \frac{1}{2} a^{9} - \frac{47}{190} a^{8} + \frac{1}{190} a^{7} + \frac{9}{95} a^{6} + \frac{13}{38} a^{5} - \frac{26}{95} a^{4} + \frac{2}{95} a^{3} - \frac{3}{38} a^{2} + \frac{42}{95} a - \frac{67}{190}$, $\frac{1}{190} a^{17} + \frac{3}{190} a^{15} + \frac{39}{190} a^{14} + \frac{3}{19} a^{13} - \frac{18}{95} a^{12} - \frac{9}{95} a^{11} + \frac{3}{19} a^{10} + \frac{24}{95} a^{9} + \frac{28}{95} a^{8} + \frac{33}{190} a^{7} + \frac{5}{19} a^{6} - \frac{27}{190} a^{5} + \frac{79}{190} a^{4} + \frac{9}{38} a^{3} - \frac{23}{95} a^{2} + \frac{53}{190} a - \frac{11}{38}$, $\frac{1}{59090} a^{18} - \frac{23}{29545} a^{17} - \frac{17}{11818} a^{16} + \frac{14711}{59090} a^{15} - \frac{11853}{59090} a^{14} + \frac{11017}{59090} a^{13} + \frac{1869}{29545} a^{12} + \frac{52}{1555} a^{11} - \frac{2274}{29545} a^{10} - \frac{16117}{59090} a^{9} - \frac{13037}{59090} a^{8} + \frac{9179}{59090} a^{7} + \frac{10719}{59090} a^{6} - \frac{28339}{59090} a^{5} + \frac{1301}{29545} a^{4} + \frac{11971}{29545} a^{3} + \frac{22869}{59090} a^{2} - \frac{485}{5909} a - \frac{19979}{59090}$, $\frac{1}{165096392226449511121026418831624987514914318935813060960097115830} a^{19} + \frac{455462026008782546568382075523581808416565508417503073557834}{82548196113224755560513209415812493757457159467906530480048557915} a^{18} + \frac{204795505092062211600902874595367531781362433884292476693862}{334203223130464597410984653505313739908733439141321985749184445} a^{17} + \frac{207653243477173083748808626441207156424111755305234766139294332}{82548196113224755560513209415812493757457159467906530480048557915} a^{16} + \frac{15608981371648129151743856517658590666151191793387181968718906}{82548196113224755560513209415812493757457159467906530480048557915} a^{15} + \frac{36243992507737585003399802114739487227191274729540151909708874817}{165096392226449511121026418831624987514914318935813060960097115830} a^{14} - \frac{11486127830729619640732880704956919679226947055779844855791913641}{82548196113224755560513209415812493757457159467906530480048557915} a^{13} - \frac{1528796194428588501259695551847501620586924209222198734125110569}{165096392226449511121026418831624987514914318935813060960097115830} a^{12} + \frac{31858498607269065552859041549840447869663292125012837083224521727}{165096392226449511121026418831624987514914318935813060960097115830} a^{11} - \frac{15955012476144880002904505452081973412370007169993913800518056739}{82548196113224755560513209415812493757457159467906530480048557915} a^{10} - \frac{2698247624942327106253766274812651587915575195869294176379885909}{6349861239478827350808708416600961058265935343685117729234504455} a^{9} - \frac{72083581953454479502490520398011569284277383541352452440495278073}{165096392226449511121026418831624987514914318935813060960097115830} a^{8} + \frac{1077763248190676296257058832453540489267690514532646875746681002}{82548196113224755560513209415812493757457159467906530480048557915} a^{7} - \frac{1810298813472709097847696739864810150400225128179244331218670061}{8689283801392079532685600991138157237627069417674371629478795570} a^{6} - \frac{1398896243049997504708459041666341832822994424517853784191859308}{6349861239478827350808708416600961058265935343685117729234504455} a^{5} - \frac{63431728429594619648058663895874406988169240110053244562797906633}{165096392226449511121026418831624987514914318935813060960097115830} a^{4} + \frac{60447618652106693018329932484160993885264909724952738995116177639}{165096392226449511121026418831624987514914318935813060960097115830} a^{3} + \frac{16690441092570297926034092435360650792600781814913611455142644762}{82548196113224755560513209415812493757457159467906530480048557915} a^{2} - \frac{58010953955974135523589331530554454281529304753949122568802949101}{165096392226449511121026418831624987514914318935813060960097115830} a + \frac{39409575325316261907975497830068550069420075377495725178517512559}{82548196113224755560513209415812493757457159467906530480048557915}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{5}\times C_{150}$, which has order $18750$ (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:  \( 4747366.99665 \) (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

\(\Q(\sqrt{817}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{-86}) \), \(\Q(\sqrt{-38}, \sqrt{-86})\), 5.5.667489.1 x5, 10.10.364007458703857.1, 10.0.277390614111813632.1 x5, 10.0.627778758253051904.1 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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.15.5$x^{10} + 14 x^{8} + 40 x^{6} - 144 x^{4} - 432 x^{2} + 33632$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.5$x^{10} + 14 x^{8} + 40 x^{6} - 144 x^{4} - 432 x^{2} + 33632$$2$$5$$15$$C_{10}$$[3]^{5}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$