↑

Properties

Label 20.0.28993864831...6409.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 271^{2}\cdot 401^{8}$
Root discriminant $33.35$
Ramified primes $3, 271, 401$
Class number $24$ (GRH)
Class group $[2, 12]$ (GRH)
Galois group 20T141

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 44, 142, 667, 907, 1857, 1042, 2589, 811, 2525, 87, 1572, -17, 621, -50, 155, -4, 17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 17*x^18 - 4*x^17 + 155*x^16 - 50*x^15 + 621*x^14 - 17*x^13 + 1572*x^12 + 87*x^11 + 2525*x^10 + 811*x^9 + 2589*x^8 + 1042*x^7 + 1857*x^6 + 907*x^5 + 667*x^4 + 142*x^3 + 44*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 17*x^18 - 4*x^17 + 155*x^16 - 50*x^15 + 621*x^14 - 17*x^13 + 1572*x^12 + 87*x^11 + 2525*x^10 + 811*x^9 + 2589*x^8 + 1042*x^7 + 1857*x^6 + 907*x^5 + 667*x^4 + 142*x^3 + 44*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 17 x^{18} - 4 x^{17} + 155 x^{16} - 50 x^{15} + 621 x^{14} - 17 x^{13} + 1572 x^{12} + 87 x^{11} + 2525 x^{10} + 811 x^{9} + 2589 x^{8} + 1042 x^{7} + 1857 x^{6} + 907 x^{5} + 667 x^{4} + 142 x^{3} + 44 x^{2} - 4 x + 1 \)

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:  \(2899386483136048131727057286409=3^{10}\cdot 271^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.35$
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:  $3, 271, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{111} a^{18} + \frac{2}{111} a^{17} + \frac{3}{37} a^{16} + \frac{11}{111} a^{14} - \frac{2}{37} a^{13} - \frac{23}{111} a^{12} - \frac{13}{111} a^{11} + \frac{1}{111} a^{10} - \frac{34}{111} a^{9} + \frac{14}{37} a^{8} - \frac{31}{111} a^{7} + \frac{17}{111} a^{6} + \frac{52}{111} a^{5} + \frac{17}{111} a^{4} + \frac{32}{111} a^{3} - \frac{32}{111} a^{2} - \frac{18}{37} a + \frac{7}{111}$, $\frac{1}{7411527900196847938334391} a^{19} + \frac{15308959810148396418986}{7411527900196847938334391} a^{18} - \frac{996442642738484465124185}{2470509300065615979444797} a^{17} + \frac{188173306030995776538117}{2470509300065615979444797} a^{16} + \frac{2658671942963793712164434}{7411527900196847938334391} a^{15} - \frac{719116910131290952230533}{2470509300065615979444797} a^{14} - \frac{424351270831570212291323}{7411527900196847938334391} a^{13} + \frac{3252787165289277274698938}{7411527900196847938334391} a^{12} + \frac{2049357214434909564277090}{7411527900196847938334391} a^{11} - \frac{671311690166169221856238}{7411527900196847938334391} a^{10} + \frac{5518163854193910145302}{2470509300065615979444797} a^{9} - \frac{2579677649250909916842658}{7411527900196847938334391} a^{8} - \frac{1135822883754262038603562}{7411527900196847938334391} a^{7} + \frac{2398642144420698387702925}{7411527900196847938334391} a^{6} - \frac{1973369333318113250904310}{7411527900196847938334391} a^{5} - \frac{3320314652836371210114763}{7411527900196847938334391} a^{4} - \frac{100995383853741132824987}{7411527900196847938334391} a^{3} - \frac{885310877482765181904317}{2470509300065615979444797} a^{2} + \frac{1872625011494413650420247}{7411527900196847938334391} a - \frac{1162209045611573064330719}{2470509300065615979444797}$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (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:  \( \frac{18922539127555365866711}{200311564870185079414443} a^{19} - \frac{10925366646177723919408}{66770521623395026471481} a^{18} + \frac{311042338376229771871582}{200311564870185079414443} a^{17} + \frac{3750681230505452246982}{66770521623395026471481} a^{16} + \frac{2904435434130755891082088}{200311564870185079414443} a^{15} - \frac{161911907547303732220940}{200311564870185079414443} a^{14} + \frac{11416881686571647310679508}{200311564870185079414443} a^{13} + \frac{939046141140336214437770}{66770521623395026471481} a^{12} + \frac{9776611148450332497703986}{66770521623395026471481} a^{11} + \frac{3168108147347071328416297}{66770521623395026471481} a^{10} + \frac{47322757435032067992584560}{200311564870185079414443} a^{9} + \frac{27796569335106492827355880}{200311564870185079414443} a^{8} + \frac{51562890219100958538472766}{200311564870185079414443} a^{7} + \frac{10668143782548758156878742}{66770521623395026471481} a^{6} + \frac{12863839153585870042556620}{66770521623395026471481} a^{5} + \frac{25570333318733335985041574}{200311564870185079414443} a^{4} + \frac{5274199427352828256805292}{66770521623395026471481} a^{3} + \frac{5438097127584772750641812}{200311564870185079414443} a^{2} + \frac{913107010202296328424254}{200311564870185079414443} a + \frac{123410157810045022530578}{200311564870185079414443} \) (order $6$)
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:  \( 1302372.67752 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.160801.1, 10.10.21021709781613.1, 10.0.567586164103551.1, 10.0.6283241669043.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
271Data not computed
401Data not computed