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Properties

Label 20.0.28269767801...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 11^{16}\cdot 17^{10}$
Root discriminant $93.88$
Ramified primes $5, 11, 17$
Class number $1040722$ (GRH)
Class group $[1040722]$ (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![43048528051, -10396170396, 29810100332, -7175480378, 9866835796, -2240309653, 2023763270, -411643979, 282578412, -49003348, 27789601, -3903468, 1971087, -206886, 101401, -6964, 3679, -132, 86, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 86*x^18 - 132*x^17 + 3679*x^16 - 6964*x^15 + 101401*x^14 - 206886*x^13 + 1971087*x^12 - 3903468*x^11 + 27789601*x^10 - 49003348*x^9 + 282578412*x^8 - 411643979*x^7 + 2023763270*x^6 - 2240309653*x^5 + 9866835796*x^4 - 7175480378*x^3 + 29810100332*x^2 - 10396170396*x + 43048528051)
 
gp: K = bnfinit(x^20 - x^19 + 86*x^18 - 132*x^17 + 3679*x^16 - 6964*x^15 + 101401*x^14 - 206886*x^13 + 1971087*x^12 - 3903468*x^11 + 27789601*x^10 - 49003348*x^9 + 282578412*x^8 - 411643979*x^7 + 2023763270*x^6 - 2240309653*x^5 + 9866835796*x^4 - 7175480378*x^3 + 29810100332*x^2 - 10396170396*x + 43048528051, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 86 x^{18} - 132 x^{17} + 3679 x^{16} - 6964 x^{15} + 101401 x^{14} - 206886 x^{13} + 1971087 x^{12} - 3903468 x^{11} + 27789601 x^{10} - 49003348 x^{9} + 282578412 x^{8} - 411643979 x^{7} + 2023763270 x^{6} - 2240309653 x^{5} + 9866835796 x^{4} - 7175480378 x^{3} + 29810100332 x^{2} - 10396170396 x + 43048528051 \)

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:  \(2826976780158713912797906502960205078125=5^{15}\cdot 11^{16}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.88$
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:  $5, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(935=5\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{935}(256,·)$, $\chi_{935}(1,·)$, $\chi_{935}(322,·)$, $\chi_{935}(67,·)$, $\chi_{935}(324,·)$, $\chi_{935}(69,·)$, $\chi_{935}(577,·)$, $\chi_{935}(713,·)$, $\chi_{935}(203,·)$, $\chi_{935}(834,·)$, $\chi_{935}(152,·)$, $\chi_{935}(851,·)$, $\chi_{935}(917,·)$, $\chi_{935}(86,·)$, $\chi_{935}(664,·)$, $\chi_{935}(543,·)$, $\chi_{935}(749,·)$, $\chi_{935}(883,·)$, $\chi_{935}(628,·)$, $\chi_{935}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{29} a^{15} + \frac{7}{29} a^{14} - \frac{1}{29} a^{13} + \frac{1}{29} a^{12} + \frac{3}{29} a^{11} - \frac{2}{29} a^{10} - \frac{14}{29} a^{9} - \frac{11}{29} a^{8} - \frac{6}{29} a^{7} - \frac{4}{29} a^{6} + \frac{9}{29} a^{5} - \frac{12}{29} a^{4} - \frac{5}{29} a^{3} + \frac{10}{29} a^{2} + \frac{1}{29} a - \frac{12}{29}$, $\frac{1}{29} a^{16} + \frac{8}{29} a^{14} + \frac{8}{29} a^{13} - \frac{4}{29} a^{12} + \frac{6}{29} a^{11} + \frac{13}{29} a^{8} + \frac{9}{29} a^{7} + \frac{8}{29} a^{6} + \frac{12}{29} a^{5} - \frac{8}{29} a^{4} - \frac{13}{29} a^{3} - \frac{11}{29} a^{2} + \frac{10}{29} a - \frac{3}{29}$, $\frac{1}{29} a^{17} + \frac{10}{29} a^{14} + \frac{4}{29} a^{13} - \frac{2}{29} a^{12} + \frac{5}{29} a^{11} - \frac{13}{29} a^{10} + \frac{9}{29} a^{9} + \frac{10}{29} a^{8} - \frac{2}{29} a^{7} - \frac{14}{29} a^{6} + \frac{7}{29} a^{5} - \frac{4}{29} a^{4} - \frac{12}{29} a^{2} - \frac{11}{29} a + \frac{9}{29}$, $\frac{1}{29} a^{18} - \frac{8}{29} a^{14} + \frac{8}{29} a^{13} - \frac{5}{29} a^{12} - \frac{14}{29} a^{11} + \frac{5}{29} a^{9} - \frac{8}{29} a^{8} - \frac{12}{29} a^{7} - \frac{11}{29} a^{6} - \frac{7}{29} a^{5} + \frac{4}{29} a^{4} + \frac{9}{29} a^{3} + \frac{5}{29} a^{2} - \frac{1}{29} a + \frac{4}{29}$, $\frac{1}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{19} + \frac{48390234446531029284434686051706144116287005173125042274068549334281492703471609}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{18} - \frac{787319002075753172093758733001911969663324535824780160289904682684724044727197793}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{17} - \frac{129557975806123910560904443999574498732440109862455243364594005272322904531494469}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{16} - \frac{858474245380436123765029285750024808593261182370613205620329975853132026611831520}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{15} - \frac{18089584908778353427392116609887390184624074493242931263884592564136879277325951027}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{14} - \frac{24132930083067897874859549752528414055254290946832548356686501691031794003082587811}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{13} + \frac{2507990880502770751337795047454500013996104554460954393735655376941955428997292530}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{12} - \frac{12220465524592846193033433827722586304369228264622236980420895992080300435592731074}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{11} - \frac{530201231701067172737431107673550649192574061359758191772083014284888611230898404}{1862916336388796815122232744969877916078133670367709914088141960903537384379340031} a^{10} - \frac{11852857018772518553211590059319447066419106317135093186984944481324151788902314704}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{9} + \frac{11078926016404815436629864417984274161663261350762949942597561898202701872571665342}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{8} + \frac{18944404934672945613754143145393634278536868385440241473106019252280304324581089470}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{7} - \frac{4139083061111205129128589172791112996264438755501485445535012148570112128236538009}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{6} - \frac{25412623911537271258229417347074895719223140827659807238411470128029886690574051889}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{5} - \frac{4229321503598213373863330911636080789276992609245483884982831805387636259485664701}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{4} - \frac{7742113997878193552310375125601816959306591874574955951274061844863455165851614644}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{3} - \frac{20246739970023617922046594686693912468960863670828367216440237058954241939132906819}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a^{2} - \frac{58443890206788172662434010794564111905112109735495602744124033291310393994128043}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899} a - \frac{17099408050197726761275266974027919314230058034103352770878308225949577494639401401}{54024573755275107638544749604126459566265876440663587508556116866202584147000860899}$

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

Class group and class number

$C_{1040722}$, which has order $1040722$ (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:  \( 140644.599182 \) (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

\(\Q(\sqrt{5}) \), 4.0.36125.1, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 $20$ $20$ R $20$ R $20$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
17Data not computed