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Properties

Label 20.0.23836200853...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{15}\cdot 31^{16}$
Root discriminant $147.52$
Ramified primes $2, 5, 31$
Class number $1712800$ (GRH)
Class group $[2, 2, 2, 10, 21410]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1828090895, -2104862820, 2477103620, -1817494360, 1227776161, -632767656, 280401978, -92990264, 25613303, -4464316, 562340, -760, 40086, -22580, 10210, -1684, 143, 44, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 + 44*x^17 + 143*x^16 - 1684*x^15 + 10210*x^14 - 22580*x^13 + 40086*x^12 - 760*x^11 + 562340*x^10 - 4464316*x^9 + 25613303*x^8 - 92990264*x^7 + 280401978*x^6 - 632767656*x^5 + 1227776161*x^4 - 1817494360*x^3 + 2477103620*x^2 - 2104862820*x + 1828090895)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 + 44*x^17 + 143*x^16 - 1684*x^15 + 10210*x^14 - 22580*x^13 + 40086*x^12 - 760*x^11 + 562340*x^10 - 4464316*x^9 + 25613303*x^8 - 92990264*x^7 + 280401978*x^6 - 632767656*x^5 + 1227776161*x^4 - 1817494360*x^3 + 2477103620*x^2 - 2104862820*x + 1828090895, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} + 44 x^{17} + 143 x^{16} - 1684 x^{15} + 10210 x^{14} - 22580 x^{13} + 40086 x^{12} - 760 x^{11} + 562340 x^{10} - 4464316 x^{9} + 25613303 x^{8} - 92990264 x^{7} + 280401978 x^{6} - 632767656 x^{5} + 1227776161 x^{4} - 1817494360 x^{3} + 2477103620 x^{2} - 2104862820 x + 1828090895 \)

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:  \(23836200853411766725131665408000000000000000=2^{30}\cdot 5^{15}\cdot 31^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $147.52$
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:  $2, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1240=2^{3}\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1240}(1,·)$, $\chi_{1240}(1093,·)$, $\chi_{1240}(1089,·)$, $\chi_{1240}(841,·)$, $\chi_{1240}(717,·)$, $\chi_{1240}(653,·)$, $\chi_{1240}(529,·)$, $\chi_{1240}(853,·)$, $\chi_{1240}(281,·)$, $\chi_{1240}(729,·)$, $\chi_{1240}(1117,·)$, $\chi_{1240}(481,·)$, $\chi_{1240}(357,·)$, $\chi_{1240}(721,·)$, $\chi_{1240}(157,·)$, $\chi_{1240}(373,·)$, $\chi_{1240}(969,·)$, $\chi_{1240}(249,·)$, $\chi_{1240}(1213,·)$, $\chi_{1240}(597,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{10} a^{16} - \frac{1}{5} a^{15} + \frac{1}{10} a^{12} + \frac{1}{10} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{5} + \frac{1}{10} a^{4}$, $\frac{1}{50} a^{17} + \frac{1}{50} a^{15} + \frac{1}{5} a^{14} + \frac{3}{25} a^{13} - \frac{3}{50} a^{12} + \frac{1}{50} a^{11} - \frac{4}{25} a^{10} - \frac{9}{50} a^{9} - \frac{3}{50} a^{8} - \frac{1}{2} a^{7} - \frac{3}{50} a^{6} + \frac{1}{5} a^{5} + \frac{11}{25} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{3}{10}$, $\frac{1}{4323657680095991499871450} a^{18} + \frac{1862122265869340621736}{2161828840047995749935725} a^{17} + \frac{44547993089385254108648}{2161828840047995749935725} a^{16} + \frac{518726409633866756455396}{2161828840047995749935725} a^{15} - \frac{780895377165380015128549}{4323657680095991499871450} a^{14} + \frac{47742745389727952189077}{2161828840047995749935725} a^{13} - \frac{126346935905987486491939}{864731536019198299974290} a^{12} + \frac{970993138402982965857339}{4323657680095991499871450} a^{11} + \frac{10466860211594793353606}{432365768009599149987145} a^{10} + \frac{601551262562399157373399}{4323657680095991499871450} a^{9} - \frac{643635139656099142230123}{2161828840047995749935725} a^{8} + \frac{371120241667500606561761}{2161828840047995749935725} a^{7} + \frac{570739158631211639366219}{4323657680095991499871450} a^{6} + \frac{1374751988726604285686257}{4323657680095991499871450} a^{5} - \frac{36644274604015343492968}{2161828840047995749935725} a^{4} + \frac{128923304108765415961361}{864731536019198299974290} a^{3} + \frac{47610675363146061798132}{432365768009599149987145} a^{2} + \frac{47821752943897841456633}{864731536019198299974290} a - \frac{82409303685330962122902}{432365768009599149987145}$, $\frac{1}{109576054752442892149178761346640321069856936180358050} a^{19} + \frac{2349756242721277920924230881}{21915210950488578429835752269328064213971387236071610} a^{18} + \frac{204299002783605314206277489754650526958242102427017}{109576054752442892149178761346640321069856936180358050} a^{17} + \frac{179059515589377776791213292740476869719553493612029}{21915210950488578429835752269328064213971387236071610} a^{16} - \frac{3171654251795186831565628046660131757395568145195569}{54788027376221446074589380673320160534928468090179025} a^{15} - \frac{5472677976354651613148995844045403013480713843497113}{109576054752442892149178761346640321069856936180358050} a^{14} - \frac{2682818047649349308845327816931055398501953077583109}{54788027376221446074589380673320160534928468090179025} a^{13} + \frac{8218715453577845717287479936963318172869259894515729}{109576054752442892149178761346640321069856936180358050} a^{12} + \frac{9719594196031430490300575335059093310778831458884096}{54788027376221446074589380673320160534928468090179025} a^{11} + \frac{13365679803347470884079826720875634445406754437483677}{54788027376221446074589380673320160534928468090179025} a^{10} - \frac{53272702373101267322868058168172044624141318653483609}{109576054752442892149178761346640321069856936180358050} a^{9} + \frac{15286783297179695992817037704677206869473373492170377}{54788027376221446074589380673320160534928468090179025} a^{8} - \frac{6455329201549016212000552770860879238457611535551851}{21915210950488578429835752269328064213971387236071610} a^{7} - \frac{23206411178123935425753706526033788122033891657128038}{54788027376221446074589380673320160534928468090179025} a^{6} + \frac{7435677069009254955920379956420052841526501005757753}{21915210950488578429835752269328064213971387236071610} a^{5} - \frac{21528514379107142525387577815482952790390370885064529}{54788027376221446074589380673320160534928468090179025} a^{4} - \frac{1815436993048245543343002165269003784299434959835953}{21915210950488578429835752269328064213971387236071610} a^{3} + \frac{848739824529248731513156301697195993653516597303883}{4383042190097715685967150453865612842794277447214322} a^{2} + \frac{943672861618350973442031568452188618898844907932600}{2191521095048857842983575226932806421397138723607161} a + \frac{4493601439862208894115091855745594340408674045141659}{10957605475244289214917876134664032106985693618035805}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{21410}$, which has order $1712800$ (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:  \( 24173706.832424585 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.8000.2, 5.5.923521.1, 10.10.2665284492003125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$31$31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$