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Properties

Label 20.0.15850669370...4821.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{19}\cdot 223^{9}$
Root discriminant $32.36$
Ramified primes $3, 223$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![600301, 1358483, 177178, -2238513, -209409, 1558050, 36765, -628017, 30141, 135932, 6799, -22606, -7719, 5046, 1788, -1356, 204, -30, 31, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 31*x^18 - 30*x^17 + 204*x^16 - 1356*x^15 + 1788*x^14 + 5046*x^13 - 7719*x^12 - 22606*x^11 + 6799*x^10 + 135932*x^9 + 30141*x^8 - 628017*x^7 + 36765*x^6 + 1558050*x^5 - 209409*x^4 - 2238513*x^3 + 177178*x^2 + 1358483*x + 600301)
 
gp: K = bnfinit(x^20 - 10*x^19 + 31*x^18 - 30*x^17 + 204*x^16 - 1356*x^15 + 1788*x^14 + 5046*x^13 - 7719*x^12 - 22606*x^11 + 6799*x^10 + 135932*x^9 + 30141*x^8 - 628017*x^7 + 36765*x^6 + 1558050*x^5 - 209409*x^4 - 2238513*x^3 + 177178*x^2 + 1358483*x + 600301, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 31 x^{18} - 30 x^{17} + 204 x^{16} - 1356 x^{15} + 1788 x^{14} + 5046 x^{13} - 7719 x^{12} - 22606 x^{11} + 6799 x^{10} + 135932 x^{9} + 30141 x^{8} - 628017 x^{7} + 36765 x^{6} + 1558050 x^{5} - 209409 x^{4} - 2238513 x^{3} + 177178 x^{2} + 1358483 x + 600301 \)

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:  \(1585066937048382272465996314821=3^{19}\cdot 223^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.36$
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:  $3, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{9067079757758914156678073428559628487413940599785324574161} a^{19} - \frac{299650380197785695430629782692863929430618732021412830992}{9067079757758914156678073428559628487413940599785324574161} a^{18} - \frac{4337683372822017049739586528816552272425522888740055985197}{9067079757758914156678073428559628487413940599785324574161} a^{17} + \frac{224948063296759287590852161583584955572895802583486128108}{9067079757758914156678073428559628487413940599785324574161} a^{16} - \frac{3861620865992061761742318185772389897368712152976072467181}{9067079757758914156678073428559628487413940599785324574161} a^{15} + \frac{2210778024628356037100532754648982184441631665685719910637}{9067079757758914156678073428559628487413940599785324574161} a^{14} - \frac{849768226339526200806592557264983099324228309444305860242}{9067079757758914156678073428559628487413940599785324574161} a^{13} + \frac{4074215091838709372814591877602915125908210955844861454677}{9067079757758914156678073428559628487413940599785324574161} a^{12} + \frac{3968874915066284183358050396082826064695778826744606833080}{9067079757758914156678073428559628487413940599785324574161} a^{11} - \frac{4142262718659492893889333407047399208943436757800490081530}{9067079757758914156678073428559628487413940599785324574161} a^{10} + \frac{1764671471878650573127239109293814071287726452563670307927}{9067079757758914156678073428559628487413940599785324574161} a^{9} - \frac{2751999723627495756638119777535457461253121329141751023506}{9067079757758914156678073428559628487413940599785324574161} a^{8} - \frac{785210992051690799477932883522757971442889038242167007898}{9067079757758914156678073428559628487413940599785324574161} a^{7} + \frac{2624051236528283343388370692134671058121828767438209409407}{9067079757758914156678073428559628487413940599785324574161} a^{6} - \frac{454693899093681538680580933822917347385432375613943306774}{9067079757758914156678073428559628487413940599785324574161} a^{5} + \frac{3296030377954076900765909883714014117238785892356197852356}{9067079757758914156678073428559628487413940599785324574161} a^{4} + \frac{1460877235125902333534423613595850019352567278377421935618}{9067079757758914156678073428559628487413940599785324574161} a^{3} + \frac{1831946235480237820644400710477129807008264428002500938420}{9067079757758914156678073428559628487413940599785324574161} a^{2} + \frac{3319450241764935524230933896923597842148218652079154639400}{9067079757758914156678073428559628487413940599785324574161} a - \frac{2386944156452256640817893071752420683116053611604811533441}{9067079757758914156678073428559628487413940599785324574161}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{4880358174133004116830764779}{6989410514929864024320148182317537} a^{19} - \frac{82859304525204103263597893555}{6989410514929864024320148182317537} a^{18} + \frac{805595880087105257565099438703}{6989410514929864024320148182317537} a^{17} - \frac{1197282373855961483583621780224}{6989410514929864024320148182317537} a^{16} - \frac{1182314220882600639029387195099}{6989410514929864024320148182317537} a^{15} - \frac{22642557906006025596033538118635}{6989410514929864024320148182317537} a^{14} + \frac{94545319633323572714550087869248}{6989410514929864024320148182317537} a^{13} + \frac{50235542365846446009501364414160}{6989410514929864024320148182317537} a^{12} - \frac{425495112939040519117701725236249}{6989410514929864024320148182317537} a^{11} - \frac{541338539911425389708148260169352}{6989410514929864024320148182317537} a^{10} + \frac{1343718965213420927404148742335394}{6989410514929864024320148182317537} a^{9} + \frac{4419083343649266324292831507814914}{6989410514929864024320148182317537} a^{8} - \frac{3754933806303076958445254331765554}{6989410514929864024320148182317537} a^{7} - \frac{19288295703280382813845878553486232}{6989410514929864024320148182317537} a^{6} + \frac{8038221490345672645032820459960853}{6989410514929864024320148182317537} a^{5} + \frac{43188815550034423897525931715998347}{6989410514929864024320148182317537} a^{4} - \frac{3430890616151938127777887847683657}{6989410514929864024320148182317537} a^{3} - \frac{50817776775309797217046957874961994}{6989410514929864024320148182317537} a^{2} - \frac{16697819894217950784510856726399986}{6989410514929864024320148182317537} a + \frac{4519061164537380311163324807069738}{6989410514929864024320148182317537} \) (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:  \( 10213662.9228 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.3.18063.1, 10.0.978815907.1

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 $20$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
3Data not computed
223Data not computed