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Properties

Label 20.0.10932130566...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 7^{15}\cdot 11^{9}$
Root discriminant $28.31$
Ramified primes $5, 7, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2233351, 3387292, 3668677, -59794, -613169, -440055, -141961, 69081, 83612, -25589, 4413, -7107, 1303, 294, 410, -228, 52, -12, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 12*x^17 + 52*x^16 - 228*x^15 + 410*x^14 + 294*x^13 + 1303*x^12 - 7107*x^11 + 4413*x^10 - 25589*x^9 + 83612*x^8 + 69081*x^7 - 141961*x^6 - 440055*x^5 - 613169*x^4 - 59794*x^3 + 3668677*x^2 + 3387292*x + 2233351)
 
gp: K = bnfinit(x^20 - 2*x^19 - 12*x^17 + 52*x^16 - 228*x^15 + 410*x^14 + 294*x^13 + 1303*x^12 - 7107*x^11 + 4413*x^10 - 25589*x^9 + 83612*x^8 + 69081*x^7 - 141961*x^6 - 440055*x^5 - 613169*x^4 - 59794*x^3 + 3668677*x^2 + 3387292*x + 2233351, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 12 x^{17} + 52 x^{16} - 228 x^{15} + 410 x^{14} + 294 x^{13} + 1303 x^{12} - 7107 x^{11} + 4413 x^{10} - 25589 x^{9} + 83612 x^{8} + 69081 x^{7} - 141961 x^{6} - 440055 x^{5} - 613169 x^{4} - 59794 x^{3} + 3668677 x^{2} + 3387292 x + 2233351 \)

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:  \(109321305666509476480595703125=5^{10}\cdot 7^{15}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.31$
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:  $5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{10341490957011044062937283676597717089267031301811300456797170175963} a^{19} - \frac{3783753672189291553543793764929742008099999328580979007686119709479}{10341490957011044062937283676597717089267031301811300456797170175963} a^{18} + \frac{1110062288503350011425264141567981251046451685122876075654553900100}{10341490957011044062937283676597717089267031301811300456797170175963} a^{17} + \frac{3197401024421173144218533627129979110427752153733698160871893860991}{10341490957011044062937283676597717089267031301811300456797170175963} a^{16} - \frac{4695219870647445295601062225520601047516206241710556875791796909944}{10341490957011044062937283676597717089267031301811300456797170175963} a^{15} + \frac{1030831877441447543770457991888220100476912190348689411218697285762}{10341490957011044062937283676597717089267031301811300456797170175963} a^{14} - \frac{1170789216189708115928937319158723913876593256214483362582303682022}{10341490957011044062937283676597717089267031301811300456797170175963} a^{13} + \frac{4524156484961301149835237040797989817663765460289942141890503197151}{10341490957011044062937283676597717089267031301811300456797170175963} a^{12} + \frac{1667649785210973538989141590733495404080164748922806200857772480800}{10341490957011044062937283676597717089267031301811300456797170175963} a^{11} - \frac{2027139177371325525089751912201300548729845316459158345754399449016}{10341490957011044062937283676597717089267031301811300456797170175963} a^{10} + \frac{3769622382732964327320871286424477487930024179387279283490406267434}{10341490957011044062937283676597717089267031301811300456797170175963} a^{9} + \frac{1997888306699890282139487018209152426938730654085112887310875426296}{10341490957011044062937283676597717089267031301811300456797170175963} a^{8} + \frac{3573053080543867998278332651683156138465720788341362790367528021097}{10341490957011044062937283676597717089267031301811300456797170175963} a^{7} + \frac{3518200096227944348094532465996055824585929215539062810697199567066}{10341490957011044062937283676597717089267031301811300456797170175963} a^{6} + \frac{1253720151644561135019401136937029803706854011402503929035319772899}{10341490957011044062937283676597717089267031301811300456797170175963} a^{5} - \frac{4839264985999455224485870075384617606639486812766515836745201036252}{10341490957011044062937283676597717089267031301811300456797170175963} a^{4} - \frac{3829042713244380828855770337429050929368948519894905738159036979726}{10341490957011044062937283676597717089267031301811300456797170175963} a^{3} - \frac{1834794095854624878421821664709732718389213096143118331017555679397}{10341490957011044062937283676597717089267031301811300456797170175963} a^{2} + \frac{3527029495567783480914490190698589311818896456674988495779483372726}{10341490957011044062937283676597717089267031301811300456797170175963} a - \frac{2383312896333394930216447005709532962041830009340199476294633542817}{10341490957011044062937283676597717089267031301811300456797170175963}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 623585.5305503157 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.94325.1, 10.0.246071287.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ $20$ R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ $20$

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
5Data not computed
7Data not computed
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$