↑

Properties

Label 20.4.26532206880...9904.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{50}\cdot 31^{6}\cdot 227^{4}$
Root discriminant $46.90$
Ramified primes $2, 31, 227$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1025

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6712, 18368, 15616, -1856, 70364, -37696, 5640, 61488, 19470, -21592, -11312, 4104, 2261, -380, -634, -64, 57, 24, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 6*x^18 + 24*x^17 + 57*x^16 - 64*x^15 - 634*x^14 - 380*x^13 + 2261*x^12 + 4104*x^11 - 11312*x^10 - 21592*x^9 + 19470*x^8 + 61488*x^7 + 5640*x^6 - 37696*x^5 + 70364*x^4 - 1856*x^3 + 15616*x^2 + 18368*x + 6712)
 
gp: K = bnfinit(x^20 - 4*x^19 - 6*x^18 + 24*x^17 + 57*x^16 - 64*x^15 - 634*x^14 - 380*x^13 + 2261*x^12 + 4104*x^11 - 11312*x^10 - 21592*x^9 + 19470*x^8 + 61488*x^7 + 5640*x^6 - 37696*x^5 + 70364*x^4 - 1856*x^3 + 15616*x^2 + 18368*x + 6712, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 6 x^{18} + 24 x^{17} + 57 x^{16} - 64 x^{15} - 634 x^{14} - 380 x^{13} + 2261 x^{12} + 4104 x^{11} - 11312 x^{10} - 21592 x^{9} + 19470 x^{8} + 61488 x^{7} + 5640 x^{6} - 37696 x^{5} + 70364 x^{4} - 1856 x^{3} + 15616 x^{2} + 18368 x + 6712 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2653220688038813933780612524539904=2^{50}\cdot 31^{6}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.90$
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:  $2, 31, 227$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{13} - \frac{1}{2} a^{10} - \frac{1}{8} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{48} a^{18} - \frac{1}{24} a^{16} + \frac{1}{12} a^{15} - \frac{1}{48} a^{14} - \frac{1}{12} a^{13} + \frac{1}{8} a^{12} - \frac{1}{48} a^{10} - \frac{1}{12} a^{9} - \frac{1}{3} a^{8} - \frac{1}{4} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{12} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{1254036452963551636207273476107453783691184897296} a^{19} - \frac{8082332560543688925020880260770954980947104409}{1254036452963551636207273476107453783691184897296} a^{18} + \frac{6058106510585286381671900792774210765166893087}{627018226481775818103636738053726891845592448648} a^{17} - \frac{32539659764424758499431921713773985493594291255}{627018226481775818103636738053726891845592448648} a^{16} + \frac{3784382764997987825392585752774342141159072477}{32154780845219272723263422464293686761312433264} a^{15} + \frac{108326882504271001249127057653715685903738006457}{1254036452963551636207273476107453783691184897296} a^{14} - \frac{64899854127529518686396077875710128627720286447}{627018226481775818103636738053726891845592448648} a^{13} + \frac{24954331357427623256226105090400681963448597269}{209006075493925272701212246017908963948530816216} a^{12} - \frac{6913093383022319089221908016623255356167706021}{96464342535657818169790267392881060283937299792} a^{11} - \frac{1853258041810963415655773202366275902362522167}{1254036452963551636207273476107453783691184897296} a^{10} - \frac{78853263675952995638633791929548305905098756621}{313509113240887909051818369026863445922796224324} a^{9} - \frac{42403330557885150754512227070558791277754188665}{156754556620443954525909184513431722961398112162} a^{8} + \frac{142764870024736175363831837416244173306354895807}{313509113240887909051818369026863445922796224324} a^{7} - \frac{154476124783450864862972446482352920422623815821}{313509113240887909051818369026863445922796224324} a^{6} - \frac{5224527865328459432529415993561877409287947081}{26125759436740659087651530752238620493566352027} a^{5} + \frac{12238124376049130019870764407454238405789468927}{52251518873481318175303061504477240987132704054} a^{4} - \frac{55911624120147240945403214361574667666397062267}{313509113240887909051818369026863445922796224324} a^{3} - \frac{90095318711229370640319803386112866461224744333}{313509113240887909051818369026863445922796224324} a^{2} - \frac{9176036829311257034807835173549303935760060489}{78377278310221977262954592256715861480699056081} a + \frac{22523561233093227197326254815047354777912527617}{78377278310221977262954592256715861480699056081}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 636682362.202 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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

Sibling fields

Degree 20 sibling: 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 $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.12.28.57$x^{12} + 2 x^{10} - 2 x^{8} + 4 x^{5} - 2 x^{4} - 2$$12$$1$$28$12T48$[2, 8/3, 8/3, 3]_{3}^{2}$
31Data not computed
227Data not computed