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Properties

Label 20.0.28112486951...2176.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 401^{9}$
Root discriminant $29.68$
Ramified primes $2, 401$
Class number $5$
Class group $[5]$
Galois group $C_5:D_4$ (as 20T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1261, -2000, -499, 4204, 6096, -15410, 13244, -164, -7787, 6486, -399, -2802, 2195, -382, -387, 272, -19, -52, 31, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 31*x^18 - 52*x^17 - 19*x^16 + 272*x^15 - 387*x^14 - 382*x^13 + 2195*x^12 - 2802*x^11 - 399*x^10 + 6486*x^9 - 7787*x^8 - 164*x^7 + 13244*x^6 - 15410*x^5 + 6096*x^4 + 4204*x^3 - 499*x^2 - 2000*x + 1261)
 
gp: K = bnfinit(x^20 - 8*x^19 + 31*x^18 - 52*x^17 - 19*x^16 + 272*x^15 - 387*x^14 - 382*x^13 + 2195*x^12 - 2802*x^11 - 399*x^10 + 6486*x^9 - 7787*x^8 - 164*x^7 + 13244*x^6 - 15410*x^5 + 6096*x^4 + 4204*x^3 - 499*x^2 - 2000*x + 1261, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 31 x^{18} - 52 x^{17} - 19 x^{16} + 272 x^{15} - 387 x^{14} - 382 x^{13} + 2195 x^{12} - 2802 x^{11} - 399 x^{10} + 6486 x^{9} - 7787 x^{8} - 164 x^{7} + 13244 x^{6} - 15410 x^{5} + 6096 x^{4} + 4204 x^{3} - 499 x^{2} - 2000 x + 1261 \)

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:  \(281124869512857892173749682176=2^{20}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.68$
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:  $2, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{117721548251110627537130675021055909} a^{19} + \frac{16343568391056094337297394211701301}{117721548251110627537130675021055909} a^{18} + \frac{10259752713598454891051339542031237}{117721548251110627537130675021055909} a^{17} + \frac{40300867988443320638374614932596742}{117721548251110627537130675021055909} a^{16} + \frac{6876759115148502657806888091578911}{117721548251110627537130675021055909} a^{15} - \frac{8621026240845915208326443976094396}{117721548251110627537130675021055909} a^{14} + \frac{50400507125510512132243600611510637}{117721548251110627537130675021055909} a^{13} - \frac{10644423680513360694738259907223434}{39240516083703542512376891673685303} a^{12} + \frac{3381947122603067650345807835877815}{117721548251110627537130675021055909} a^{11} - \frac{3361687779723736501984294520687381}{39240516083703542512376891673685303} a^{10} + \frac{52437160677729533937795765476765936}{117721548251110627537130675021055909} a^{9} - \frac{51913865549437304124033068225762219}{117721548251110627537130675021055909} a^{8} + \frac{57805393239789913497770531153272079}{117721548251110627537130675021055909} a^{7} - \frac{55213199684707665657240720769830986}{117721548251110627537130675021055909} a^{6} - \frac{47926746619337992970718409094844866}{117721548251110627537130675021055909} a^{5} - \frac{14473878000015920235822766956646342}{39240516083703542512376891673685303} a^{4} - \frac{18867471203307383236729893147673150}{117721548251110627537130675021055909} a^{3} + \frac{14235549383172770722715134390216466}{117721548251110627537130675021055909} a^{2} + \frac{4595333237891517636987379152133201}{39240516083703542512376891673685303} a + \frac{35281732782083077875372462575821379}{117721548251110627537130675021055909}$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( \frac{526521955045469462}{3288699282057837094567} a^{19} - \frac{1348498056532705413}{3288699282057837094567} a^{18} - \frac{5774044873645154710}{3288699282057837094567} a^{17} + \frac{47019127203922176600}{3288699282057837094567} a^{16} - \frac{93349826685512353097}{3288699282057837094567} a^{15} - \frac{46655033616902389566}{3288699282057837094567} a^{14} + \frac{504224803733019054997}{3288699282057837094567} a^{13} - \frac{582303009834533002037}{3288699282057837094567} a^{12} - \frac{966192113759013003708}{3288699282057837094567} a^{11} + \frac{3284845993886648955566}{3288699282057837094567} a^{10} - \frac{2668290351712220550109}{3288699282057837094567} a^{9} - \frac{3223262477016992183541}{3288699282057837094567} a^{8} + \frac{8634135741473797278491}{3288699282057837094567} a^{7} - \frac{6225038727252858716462}{3288699282057837094567} a^{6} - \frac{7467784252735428321481}{3288699282057837094567} a^{5} + \frac{15306774731749334461063}{3288699282057837094567} a^{4} - \frac{10231396103422571597183}{3288699282057837094567} a^{3} - \frac{4361494375566779909384}{3288699282057837094567} a^{2} - \frac{2828763493194151284127}{3288699282057837094567} a + \frac{1596240156843238032362}{3288699282057837094567} \) (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:  \( 951859.395024 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_4$ (as 20T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.6416.1, 5.5.160801.1, 10.0.26477528679424.1

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

Sibling fields

Galois closure: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
2Data not computed
401Data not computed