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Properties

Label 20.0.75263294618...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{15}\cdot 11^{16}\cdot 13^{15}$
Root discriminant $311.77$
Ramified primes $2, 5, 11, 13$
Class number $558743072$ (GRH)
Class group $[2, 4, 69842884]$ (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![2086214435804201, -231449391531422, 567642204410037, -56512538407374, 70722381249776, -6267755680692, 5311831568589, -414258891012, 266292849854, -17976765812, 9309194200, -531124316, 229810454, -10685780, 3956104, -141244, 45460, -1114, 315, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 315*x^18 - 1114*x^17 + 45460*x^16 - 141244*x^15 + 3956104*x^14 - 10685780*x^13 + 229810454*x^12 - 531124316*x^11 + 9309194200*x^10 - 17976765812*x^9 + 266292849854*x^8 - 414258891012*x^7 + 5311831568589*x^6 - 6267755680692*x^5 + 70722381249776*x^4 - 56512538407374*x^3 + 567642204410037*x^2 - 231449391531422*x + 2086214435804201)
 
gp: K = bnfinit(x^20 - 4*x^19 + 315*x^18 - 1114*x^17 + 45460*x^16 - 141244*x^15 + 3956104*x^14 - 10685780*x^13 + 229810454*x^12 - 531124316*x^11 + 9309194200*x^10 - 17976765812*x^9 + 266292849854*x^8 - 414258891012*x^7 + 5311831568589*x^6 - 6267755680692*x^5 + 70722381249776*x^4 - 56512538407374*x^3 + 567642204410037*x^2 - 231449391531422*x + 2086214435804201, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 315 x^{18} - 1114 x^{17} + 45460 x^{16} - 141244 x^{15} + 3956104 x^{14} - 10685780 x^{13} + 229810454 x^{12} - 531124316 x^{11} + 9309194200 x^{10} - 17976765812 x^{9} + 266292849854 x^{8} - 414258891012 x^{7} + 5311831568589 x^{6} - 6267755680692 x^{5} + 70722381249776 x^{4} - 56512538407374 x^{3} + 567642204410037 x^{2} - 231449391531422 x + 2086214435804201 \)

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:  \(75263294618341622670336367123708064000000000000000=2^{20}\cdot 5^{15}\cdot 11^{16}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $311.77$
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:  $2, 5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2860=2^{2}\cdot 5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2860}(1,·)$, $\chi_{2860}(707,·)$, $\chi_{2860}(389,·)$, $\chi_{2860}(521,·)$, $\chi_{2860}(203,·)$, $\chi_{2860}(463,·)$, $\chi_{2860}(1169,·)$, $\chi_{2860}(1747,·)$, $\chi_{2860}(1301,·)$, $\chi_{2860}(2787,·)$, $\chi_{2860}(983,·)$, $\chi_{2860}(2267,·)$, $\chi_{2860}(2341,·)$, $\chi_{2860}(2209,·)$, $\chi_{2860}(1763,·)$, $\chi_{2860}(2469,·)$, $\chi_{2860}(2729,·)$, $\chi_{2860}(2803,·)$, $\chi_{2860}(2007,·)$, $\chi_{2860}(2601,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{13} + \frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{1552766010821371901904645124} a^{18} - \frac{62501324641959326722270625}{776383005410685950952322562} a^{17} + \frac{19953898199763882581054825}{776383005410685950952322562} a^{16} - \frac{18518982563841568692914745}{1552766010821371901904645124} a^{15} + \frac{598877657783425416942383487}{1552766010821371901904645124} a^{14} + \frac{698003395256681192393763027}{1552766010821371901904645124} a^{13} - \frac{177948343054614680598243008}{388191502705342975476161281} a^{12} - \frac{185578522148089790115932483}{388191502705342975476161281} a^{11} - \frac{369631454620153901604848245}{1552766010821371901904645124} a^{10} + \frac{83609488523527425684670643}{1552766010821371901904645124} a^{9} + \frac{2030345429547722064509787}{388191502705342975476161281} a^{8} + \frac{160453415889934610296948965}{1552766010821371901904645124} a^{7} + \frac{151477408966377546340164309}{1552766010821371901904645124} a^{6} + \frac{712409745128444790310397465}{1552766010821371901904645124} a^{5} - \frac{83377711680458802206048598}{388191502705342975476161281} a^{4} + \frac{200985114518724480190492717}{1552766010821371901904645124} a^{3} + \frac{378993081909977627255867405}{1552766010821371901904645124} a^{2} - \frac{159270399714616029510560640}{388191502705342975476161281} a + \frac{422580395904002737960394177}{1552766010821371901904645124}$, $\frac{1}{4922861560801078917646335052561406164651894954302803284625299396194484} a^{19} - \frac{1202313815652552740353890758997344967548263}{4922861560801078917646335052561406164651894954302803284625299396194484} a^{18} + \frac{306972032294509594957791734602405077215827907374533965217404382208529}{2461430780400539458823167526280703082325947477151401642312649698097242} a^{17} - \frac{579537653455192960746791609058506190595833790674648508172380699074161}{4922861560801078917646335052561406164651894954302803284625299396194484} a^{16} - \frac{111402415641329683839362777672832700155110908952332352298790114333189}{1230715390200269729411583763140351541162973738575700821156324849048621} a^{15} - \frac{1045260302293971464801353739775750732298084876911273033477062722732299}{2461430780400539458823167526280703082325947477151401642312649698097242} a^{14} + \frac{1423533342317190528746755391602035364097509970403691259535834580328275}{4922861560801078917646335052561406164651894954302803284625299396194484} a^{13} + \frac{240068812873528429687322866013083559839670680575366578795993343133381}{2461430780400539458823167526280703082325947477151401642312649698097242} a^{12} - \frac{1443946114546923251215603695571305633608049343585267901134047353481971}{4922861560801078917646335052561406164651894954302803284625299396194484} a^{11} - \frac{1041105370197316624791099520043264605138804450981013148203816574184043}{2461430780400539458823167526280703082325947477151401642312649698097242} a^{10} + \frac{1144848442718624522736369477905610674847516431123520003185638056867525}{4922861560801078917646335052561406164651894954302803284625299396194484} a^{9} + \frac{2427495395822737706517118599950160651371061843645048151873245247771987}{4922861560801078917646335052561406164651894954302803284625299396194484} a^{8} + \frac{426747318028166776859866898932885026507524740207099114523361359950675}{1230715390200269729411583763140351541162973738575700821156324849048621} a^{7} + \frac{447368413983576418876086747972798832979080069115548453704916283126180}{1230715390200269729411583763140351541162973738575700821156324849048621} a^{6} + \frac{936047244853023506206176633544937912289168285423130380521215670464087}{4922861560801078917646335052561406164651894954302803284625299396194484} a^{5} - \frac{1735316473431945514618939603051726967892292406003710150099191096667793}{4922861560801078917646335052561406164651894954302803284625299396194484} a^{4} + \frac{18419344690630600290451494295160925569858026263003148867684328156091}{2461430780400539458823167526280703082325947477151401642312649698097242} a^{3} - \frac{2015371232162978814240358072592457712905529520956988622647372589554683}{4922861560801078917646335052561406164651894954302803284625299396194484} a^{2} - \frac{469925723154523752119397841836186494651774617363869455499348988909995}{4922861560801078917646335052561406164651894954302803284625299396194484} a + \frac{1258170949191530836344290983339073077407863110572479245085355155696691}{4922861560801078917646335052561406164651894954302803284625299396194484}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{69842884}$, which has order $558743072$ (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:  \( 40471423.40714512 \) (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

\(\Q(\sqrt{65}) \), 4.0.4394000.2, \(\Q(\zeta_{11})^+\), 10.10.248718600009790625.1

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 ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R R $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ $20$ $20$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
5Data not computed
11Data not computed
13Data not computed