↑

Properties

Label 20.0.13124183392...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 103^{10}$
Root discriminant $22.69$
Ramified primes $5, 103$
Class number $3$
Class group $[3]$
Galois group $D_{10}$ (as 20T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -243, 1269, -2889, 6277, -11735, 20732, -28955, 35494, -38148, 36198, -28542, 18417, -9996, 4683, -1890, 649, -188, 44, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 44*x^18 - 188*x^17 + 649*x^16 - 1890*x^15 + 4683*x^14 - 9996*x^13 + 18417*x^12 - 28542*x^11 + 36198*x^10 - 38148*x^9 + 35494*x^8 - 28955*x^7 + 20732*x^6 - 11735*x^5 + 6277*x^4 - 2889*x^3 + 1269*x^2 - 243*x + 81)
 
gp: K = bnfinit(x^20 - 8*x^19 + 44*x^18 - 188*x^17 + 649*x^16 - 1890*x^15 + 4683*x^14 - 9996*x^13 + 18417*x^12 - 28542*x^11 + 36198*x^10 - 38148*x^9 + 35494*x^8 - 28955*x^7 + 20732*x^6 - 11735*x^5 + 6277*x^4 - 2889*x^3 + 1269*x^2 - 243*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 44 x^{18} - 188 x^{17} + 649 x^{16} - 1890 x^{15} + 4683 x^{14} - 9996 x^{13} + 18417 x^{12} - 28542 x^{11} + 36198 x^{10} - 38148 x^{9} + 35494 x^{8} - 28955 x^{7} + 20732 x^{6} - 11735 x^{5} + 6277 x^{4} - 2889 x^{3} + 1269 x^{2} - 243 x + 81 \)

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:  \(1312418339203244062978515625=5^{10}\cdot 103^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.69$
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:  $5, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} + \frac{4}{9} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{11} - \frac{1}{9} a^{7} - \frac{1}{3} a^{5} - \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{12} - \frac{1}{9} a^{8} - \frac{1}{3} a^{6} - \frac{1}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{27} a^{17} + \frac{1}{27} a^{14} - \frac{2}{27} a^{12} - \frac{1}{27} a^{11} - \frac{1}{27} a^{10} - \frac{1}{9} a^{9} + \frac{2}{27} a^{8} + \frac{10}{27} a^{7} + \frac{7}{27} a^{6} - \frac{11}{27} a^{5} + \frac{4}{27} a^{4} + \frac{11}{27} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{1485} a^{18} - \frac{17}{1485} a^{17} + \frac{16}{495} a^{16} + \frac{43}{1485} a^{15} + \frac{8}{297} a^{14} - \frac{7}{135} a^{13} + \frac{7}{495} a^{12} - \frac{197}{1485} a^{11} + \frac{1}{135} a^{10} - \frac{223}{1485} a^{9} - \frac{58}{495} a^{8} - \frac{4}{27} a^{7} + \frac{52}{297} a^{6} - \frac{1}{27} a^{5} + \frac{194}{495} a^{4} - \frac{8}{135} a^{3} - \frac{1}{165} a^{2} + \frac{17}{55} a + \frac{17}{55}$, $\frac{1}{4764968302960004903951500665} a^{19} + \frac{497959991202048731097472}{4764968302960004903951500665} a^{18} + \frac{10685384723002440200356049}{952993660592000980790300133} a^{17} + \frac{21768020425027973124687848}{952993660592000980790300133} a^{16} - \frac{180749445060417092515140038}{4764968302960004903951500665} a^{15} - \frac{7156824997263161803506497}{433178936632727718541045515} a^{14} - \frac{7114512609910052284529443}{529440922551111655994611185} a^{13} + \frac{657777652416520188211906457}{4764968302960004903951500665} a^{12} - \frac{4631160020643041472800077}{433178936632727718541045515} a^{11} + \frac{793167020272765816290684376}{4764968302960004903951500665} a^{10} - \frac{1576304758288039245531179}{29968354106666697509128935} a^{9} + \frac{15658467525064454680680449}{433178936632727718541045515} a^{8} - \frac{17620279613024326336376261}{35296061503407443732974079} a^{7} - \frac{4187803846260059600638597}{9626198591838393745356567} a^{6} - \frac{1777800677905217399838411173}{4764968302960004903951500665} a^{5} + \frac{4884710005299393000366031}{28878595775515181236069701} a^{4} - \frac{585135765237964230494626321}{4764968302960004903951500665} a^{3} + \frac{467604341989358696339711966}{1588322767653334967983833555} a^{2} - \frac{7684199518914606759095311}{35296061503407443732974079} a - \frac{3899211501484296540843157}{16043664319730656242260945}$

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

Class group and class number

$C_{3}$, which has order $3$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 126437.428056 \)
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

\(\Q(\sqrt{-103}) \), \(\Q(\sqrt{-515}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-103})\), 5.1.10609.1 x5, 10.0.11592740743.1, 10.0.36227314821875.1 x5, 10.2.351721503125.1 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$103$103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$