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Properties

Label 20.0.44537115538...9649.1
Degree $20$
Signature $[0, 10]$
Discriminant $23^{10}\cdot 401^{10}$
Root discriminant $96.04$
Ramified primes $23, 401$
Class number $784176$ (GRH)
Class group $[2, 2, 62, 3162]$ (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![2591170963, -2600226987, 3484263346, -2478325380, 1892748576, -1028739177, 561487303, -240286113, 100182550, -33719964, 10840035, -2760297, 658324, -111411, 17509, -1098, 210, -42, 31, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 31*x^18 - 42*x^17 + 210*x^16 - 1098*x^15 + 17509*x^14 - 111411*x^13 + 658324*x^12 - 2760297*x^11 + 10840035*x^10 - 33719964*x^9 + 100182550*x^8 - 240286113*x^7 + 561487303*x^6 - 1028739177*x^5 + 1892748576*x^4 - 2478325380*x^3 + 3484263346*x^2 - 2600226987*x + 2591170963)
 
gp: K = bnfinit(x^20 - 6*x^19 + 31*x^18 - 42*x^17 + 210*x^16 - 1098*x^15 + 17509*x^14 - 111411*x^13 + 658324*x^12 - 2760297*x^11 + 10840035*x^10 - 33719964*x^9 + 100182550*x^8 - 240286113*x^7 + 561487303*x^6 - 1028739177*x^5 + 1892748576*x^4 - 2478325380*x^3 + 3484263346*x^2 - 2600226987*x + 2591170963, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 31 x^{18} - 42 x^{17} + 210 x^{16} - 1098 x^{15} + 17509 x^{14} - 111411 x^{13} + 658324 x^{12} - 2760297 x^{11} + 10840035 x^{10} - 33719964 x^{9} + 100182550 x^{8} - 240286113 x^{7} + 561487303 x^{6} - 1028739177 x^{5} + 1892748576 x^{4} - 2478325380 x^{3} + 3484263346 x^{2} - 2600226987 x + 2591170963 \)

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:  \(4453711553843801272018640137577958609649=23^{10}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.04$
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:  $23, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{6} + \frac{2}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{7} + \frac{2}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{2}{9} a^{4} + \frac{1}{3} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{2}{9} a^{5} + \frac{1}{3} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{16} + \frac{4}{81} a^{15} - \frac{4}{81} a^{14} + \frac{1}{27} a^{13} - \frac{1}{81} a^{12} + \frac{8}{81} a^{11} - \frac{10}{81} a^{10} + \frac{7}{81} a^{9} - \frac{1}{27} a^{8} + \frac{1}{81} a^{7} - \frac{8}{81} a^{6} + \frac{38}{81} a^{5} - \frac{29}{81} a^{4} - \frac{10}{27} a^{3} + \frac{7}{81} a^{2} - \frac{38}{81} a - \frac{37}{81}$, $\frac{1}{81} a^{17} - \frac{2}{81} a^{15} + \frac{1}{81} a^{14} - \frac{4}{81} a^{13} + \frac{1}{27} a^{12} + \frac{4}{27} a^{11} - \frac{7}{81} a^{10} - \frac{13}{81} a^{9} - \frac{5}{81} a^{8} - \frac{4}{27} a^{7} - \frac{11}{81} a^{6} - \frac{10}{81} a^{5} - \frac{4}{81} a^{4} + \frac{10}{81} a^{3} - \frac{10}{27} a^{2} + \frac{34}{81} a - \frac{14}{81}$, $\frac{1}{243} a^{18} - \frac{4}{81} a^{14} - \frac{1}{27} a^{13} - \frac{8}{243} a^{12} + \frac{4}{27} a^{11} + \frac{4}{81} a^{10} - \frac{2}{27} a^{8} + \frac{2}{27} a^{7} - \frac{17}{243} a^{6} - \frac{2}{9} a^{5} + \frac{20}{81} a^{4} + \frac{7}{27} a^{3} + \frac{31}{81} a^{2} - \frac{7}{27} a + \frac{97}{243}$, $\frac{1}{39739951015292341520950118381661621669272858791986481391195691} a^{19} - \frac{108786486207841226101688059510631341268070208805944562411}{163538893067046672925720651776385274359147567045211857576937} a^{18} + \frac{39689127506636009464269254868186939209086171704527081375485}{13246650338430780506983372793887207223090952930662160463731897} a^{17} + \frac{9314836084159298650242940738512282814199137249961192223622}{13246650338430780506983372793887207223090952930662160463731897} a^{16} + \frac{154107097759800207151322308212812857166729689719740319504157}{13246650338430780506983372793887207223090952930662160463731897} a^{15} - \frac{2984730972899134684591960588477720159004838196382107919720}{4415550112810260168994457597962402407696984310220720154577299} a^{14} - \frac{1342820820418352349143478767831614579698805327674089116999751}{39739951015292341520950118381661621669272858791986481391195691} a^{13} + \frac{44371288462649336643341920089226318373316367445526667603166}{13246650338430780506983372793887207223090952930662160463731897} a^{12} - \frac{533139083065428171309431941792293405585556212975640940943181}{4415550112810260168994457597962402407696984310220720154577299} a^{11} + \frac{1628232798613082382961668528639154632666221288386549604119228}{13246650338430780506983372793887207223090952930662160463731897} a^{10} + \frac{601966406293587298892087950162303375412787014276331512818400}{4415550112810260168994457597962402407696984310220720154577299} a^{9} - \frac{1854056658917654393088404452403770831274103671713262722040024}{13246650338430780506983372793887207223090952930662160463731897} a^{8} + \frac{6461390358087296780926538450807222307443038445231795939887939}{39739951015292341520950118381661621669272858791986481391195691} a^{7} - \frac{44366924742586480120852783350497333255449034905638803387776}{13246650338430780506983372793887207223090952930662160463731897} a^{6} - \frac{334206429687127871091374392420552770236387421686259726642251}{4415550112810260168994457597962402407696984310220720154577299} a^{5} + \frac{2179999036297463326932494155936071906318325835056033573253837}{4415550112810260168994457597962402407696984310220720154577299} a^{4} - \frac{63387894692835018986860384208481386588237107433412668256103}{13246650338430780506983372793887207223090952930662160463731897} a^{3} - \frac{1781351892137484421067039841814862094240984833751522434601542}{13246650338430780506983372793887207223090952930662160463731897} a^{2} + \frac{19466832934952645399781122815381944881906897647974292524057010}{39739951015292341520950118381661621669272858791986481391195691} a + \frac{422219159556602681636378255547838694869707075510885782860849}{4415550112810260168994457597962402407696984310220720154577299}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{62}\times C_{3162}$, which has order $784176$ (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:  \( 795087.603907 \) (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

\(\Q(\sqrt{401}) \), \(\Q(\sqrt{-9223}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-23}, \sqrt{401})\), 5.5.160801.1 x5, 10.10.10368641602001.1, 10.0.66736133794547922343.1 x5, 10.0.166424273801865143.1 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed