↑

Properties

Label 20.0.22416185558...5264.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{8}\cdot 3^{10}\cdot 33769^{6}$
Root discriminant $52.18$
Ramified primes $2, 3, 33769$
Class number $648$ (GRH)
Class group $[3, 3, 72]$ (GRH)
Galois group 20T288

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 44, 1781, 6594, 19191, 29511, 40751, 32094, 33800, 18482, 19739, 5561, 6315, 355, 1608, -132, 264, -56, 33, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 33*x^18 - 56*x^17 + 264*x^16 - 132*x^15 + 1608*x^14 + 355*x^13 + 6315*x^12 + 5561*x^11 + 19739*x^10 + 18482*x^9 + 33800*x^8 + 32094*x^7 + 40751*x^6 + 29511*x^5 + 19191*x^4 + 6594*x^3 + 1781*x^2 + 44*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 33*x^18 - 56*x^17 + 264*x^16 - 132*x^15 + 1608*x^14 + 355*x^13 + 6315*x^12 + 5561*x^11 + 19739*x^10 + 18482*x^9 + 33800*x^8 + 32094*x^7 + 40751*x^6 + 29511*x^5 + 19191*x^4 + 6594*x^3 + 1781*x^2 + 44*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 33 x^{18} - 56 x^{17} + 264 x^{16} - 132 x^{15} + 1608 x^{14} + 355 x^{13} + 6315 x^{12} + 5561 x^{11} + 19739 x^{10} + 18482 x^{9} + 33800 x^{8} + 32094 x^{7} + 40751 x^{6} + 29511 x^{5} + 19191 x^{4} + 6594 x^{3} + 1781 x^{2} + 44 x + 1 \)

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:  \(22416185558603163911567862205995264=2^{8}\cdot 3^{10}\cdot 33769^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.18$
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:  $2, 3, 33769$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{7972} a^{18} + \frac{1943}{7972} a^{17} + \frac{129}{3986} a^{16} + \frac{3885}{7972} a^{15} - \frac{727}{3986} a^{14} + \frac{2761}{7972} a^{13} - \frac{592}{1993} a^{12} + \frac{313}{3986} a^{11} + \frac{2103}{7972} a^{10} - \frac{3477}{7972} a^{9} + \frac{643}{3986} a^{8} - \frac{1865}{7972} a^{7} - \frac{577}{3986} a^{6} + \frac{2201}{7972} a^{5} - \frac{3429}{7972} a^{4} - \frac{1105}{3986} a^{3} - \frac{413}{3986} a^{2} - \frac{555}{3986} a + \frac{2429}{7972}$, $\frac{1}{3811687835811936641398573021901612} a^{19} - \frac{103871142275984940620203376001}{3811687835811936641398573021901612} a^{18} + \frac{132242815941009307635162678812037}{1905843917905968320699286510950806} a^{17} + \frac{173954732883682421966108137657273}{3811687835811936641398573021901612} a^{16} - \frac{722196936811132082250642962559129}{1905843917905968320699286510950806} a^{15} - \frac{1259648714513970791339122869326667}{3811687835811936641398573021901612} a^{14} - \frac{29885719190976215271316300790745}{952921958952984160349643255475403} a^{13} + \frac{714458212578991123143346167242569}{1905843917905968320699286510950806} a^{12} + \frac{4065927889646928657084895382051}{3811687835811936641398573021901612} a^{11} + \frac{1087894514774639637468690309594047}{3811687835811936641398573021901612} a^{10} - \frac{245454146275424841539701390227275}{1905843917905968320699286510950806} a^{9} - \frac{725662514107397271198746905298829}{3811687835811936641398573021901612} a^{8} - \frac{941162361163603223265620081440323}{1905843917905968320699286510950806} a^{7} + \frac{404156326187064575433328707257977}{3811687835811936641398573021901612} a^{6} - \frac{163314714683714646862784756463425}{3811687835811936641398573021901612} a^{5} + \frac{191176435622997334109689836551321}{1905843917905968320699286510950806} a^{4} + \frac{434925464000469036480869761491653}{1905843917905968320699286510950806} a^{3} + \frac{197323163836075173059197708315035}{1905843917905968320699286510950806} a^{2} + \frac{1466836361356266532074569260435293}{3811687835811936641398573021901612} a - \frac{235841418584039091889890618535868}{952921958952984160349643255475403}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{72}$, which has order $648$ (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:  \( -\frac{48569551905713541770200495545}{1912537800206691741795571009484} a^{19} + \frac{244332412916720027960652321881}{1912537800206691741795571009484} a^{18} - \frac{402644978967003547115985352512}{478134450051672935448892752371} a^{17} + \frac{2770766017554862041123667635995}{1912537800206691741795571009484} a^{16} - \frac{6458946703171702578621608950083}{956268900103345870897785504742} a^{15} + \frac{6826249952625365064485179655637}{1912537800206691741795571009484} a^{14} - \frac{39199408159863311612647873508107}{956268900103345870897785504742} a^{13} - \frac{7391224629096244765072209803015}{956268900103345870897785504742} a^{12} - \frac{306808872521130513711234896578435}{1912537800206691741795571009484} a^{11} - \frac{260718717976897157319330729433059}{1912537800206691741795571009484} a^{10} - \frac{238222910690067438379870790654418}{478134450051672935448892752371} a^{9} - \frac{869940860274606054431729284201495}{1912537800206691741795571009484} a^{8} - \frac{810807153708523048078039452209161}{956268900103345870897785504742} a^{7} - \frac{1514345972867944977232909244613259}{1912537800206691741795571009484} a^{6} - \frac{1944504980740163627753113996116269}{1912537800206691741795571009484} a^{5} - \frac{691685588675157176768611807571727}{956268900103345870897785504742} a^{4} - \frac{225865276778598312105673341920643}{478134450051672935448892752371} a^{3} - \frac{75161372546187744210548503493162}{478134450051672935448892752371} a^{2} - \frac{84373185894742738649201920631727}{1912537800206691741795571009484} a - \frac{86001197015747151453837956449}{956268900103345870897785504742} \) (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:  \( 5520444.6153 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T288:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 36 conjugacy class representatives for t20n288
Character table for t20n288 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.135076.1, 10.0.4433662763568.1, 10.0.149720357862927792.1, 10.10.616133159929744.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
33769Data not computed