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Properties

Label 20.0.52870944306...3824.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 37^{10}$
Root discriminant $24.33$
Ramified primes $2, 37$
Class number $2$
Class group $[2]$
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![4, -64, 312, 536, 1180, -1168, 3752, -3572, 4373, -5980, 8814, -9552, 8423, -5808, 3300, -1484, 555, -164, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 38*x^18 - 164*x^17 + 555*x^16 - 1484*x^15 + 3300*x^14 - 5808*x^13 + 8423*x^12 - 9552*x^11 + 8814*x^10 - 5980*x^9 + 4373*x^8 - 3572*x^7 + 3752*x^6 - 1168*x^5 + 1180*x^4 + 536*x^3 + 312*x^2 - 64*x + 4)
 
gp: K = bnfinit(x^20 - 8*x^19 + 38*x^18 - 164*x^17 + 555*x^16 - 1484*x^15 + 3300*x^14 - 5808*x^13 + 8423*x^12 - 9552*x^11 + 8814*x^10 - 5980*x^9 + 4373*x^8 - 3572*x^7 + 3752*x^6 - 1168*x^5 + 1180*x^4 + 536*x^3 + 312*x^2 - 64*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 38 x^{18} - 164 x^{17} + 555 x^{16} - 1484 x^{15} + 3300 x^{14} - 5808 x^{13} + 8423 x^{12} - 9552 x^{11} + 8814 x^{10} - 5980 x^{9} + 4373 x^{8} - 3572 x^{7} + 3752 x^{6} - 1168 x^{5} + 1180 x^{4} + 536 x^{3} + 312 x^{2} - 64 x + 4 \)

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:  \(5287094430615384550826573824=2^{40}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.33$
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:  $2, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{136} a^{17} - \frac{1}{68} a^{16} + \frac{3}{136} a^{15} - \frac{3}{68} a^{14} + \frac{7}{136} a^{13} + \frac{15}{136} a^{12} + \frac{13}{136} a^{11} - \frac{11}{136} a^{10} - \frac{9}{68} a^{9} - \frac{25}{136} a^{8} + \frac{27}{68} a^{7} - \frac{7}{136} a^{6} + \frac{1}{17} a^{5} - \frac{19}{68} a^{4} - \frac{1}{34} a^{3} - \frac{21}{68} a^{2} + \frac{4}{17} a + \frac{15}{34}$, $\frac{1}{1904} a^{18} + \frac{33}{1904} a^{16} - \frac{1}{28} a^{15} - \frac{1}{34} a^{14} + \frac{23}{952} a^{13} - \frac{3}{136} a^{12} + \frac{33}{952} a^{11} - \frac{23}{1904} a^{10} - \frac{25}{136} a^{9} + \frac{361}{1904} a^{8} + \frac{365}{952} a^{7} + \frac{473}{952} a^{6} + \frac{139}{476} a^{5} - \frac{22}{119} a^{4} - \frac{123}{476} a^{3} - \frac{50}{119} a^{2} + \frac{75}{238} a + \frac{115}{476}$, $\frac{1}{229937952866437354235797744} a^{19} - \frac{5883216996031054682487}{229937952866437354235797744} a^{18} - \frac{6875836429991413560755}{229937952866437354235797744} a^{17} + \frac{11106645803251754238399223}{229937952866437354235797744} a^{16} + \frac{5715745337733284980230739}{114968976433218677117898872} a^{15} + \frac{5477731296441006803126453}{114968976433218677117898872} a^{14} - \frac{229595171993961861045597}{57484488216609338558949436} a^{13} - \frac{4804501202202459767240197}{57484488216609338558949436} a^{12} - \frac{10984826738128501863336229}{229937952866437354235797744} a^{11} - \frac{16999424908174334666321221}{229937952866437354235797744} a^{10} - \frac{16827949432004390348689037}{229937952866437354235797744} a^{9} + \frac{509164273097282226093351}{4892296869498667111399952} a^{8} - \frac{53994964473227107609459017}{114968976433218677117898872} a^{7} - \frac{27172456364470111166580503}{114968976433218677117898872} a^{6} - \frac{1046544907207815721159539}{28742244108304669279474718} a^{5} - \frac{33984631443086790095293}{14371122054152334639737359} a^{4} - \frac{4445299568713573903535389}{28742244108304669279474718} a^{3} + \frac{1578829394284392674971801}{28742244108304669279474718} a^{2} - \frac{913897635763425716757355}{3381440483329961091702908} a - \frac{6352109871792672294821905}{57484488216609338558949436}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{34330567450303}{788169349744091} a^{19} - \frac{154959197590283}{450382485568052} a^{18} + \frac{2552470864763247}{1576338699488182} a^{17} - \frac{21975313950494949}{3152677398976364} a^{16} + \frac{10550267318818785}{450382485568052} a^{15} - \frac{48926731148338772}{788169349744091} a^{14} + \frac{61612280859457139}{450382485568052} a^{13} - \frac{187076864060982804}{788169349744091} a^{12} + \frac{1068168401895602933}{3152677398976364} a^{11} - \frac{168632228684092765}{450382485568052} a^{10} + \frac{1055681154980611323}{3152677398976364} a^{9} - \frac{14283309874633519}{67078242531412} a^{8} + \frac{244248994797265825}{1576338699488182} a^{7} - \frac{101412069048054498}{788169349744091} a^{6} + \frac{111663718202127172}{788169349744091} a^{5} - \frac{47773677670398441}{1576338699488182} a^{4} + \frac{34359932211514354}{788169349744091} a^{3} + \frac{1374013126677575}{46362902926123} a^{2} + \frac{11407456269653903}{788169349744091} a - \frac{8928489512275}{6623271846589} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9117730.40533 \)
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

\(\Q(\sqrt{-74}) \), \(\Q(\sqrt{74}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{74})\), 5.1.87616.1 x5, 10.0.2272262782976.1, 10.2.36356204527616.2 x5, 10.0.491300061184.2 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$