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Properties

Label 20.0.10675123826...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 3^{10}\cdot 5^{22}\cdot 7^{10}$
Root discriminant $71.03$
Ramified primes $2, 3, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![103753782505, 224116200, 3006968580, 5598132760, 11025, -1433680, 86410940, -840, -46190, -629200, -351784, 880, 12310, 7400, 0, -8, 305, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 305*x^16 - 8*x^15 + 7400*x^13 + 12310*x^12 + 880*x^11 - 351784*x^10 - 629200*x^9 - 46190*x^8 - 840*x^7 + 86410940*x^6 - 1433680*x^5 + 11025*x^4 + 5598132760*x^3 + 3006968580*x^2 + 224116200*x + 103753782505)
 
gp: K = bnfinit(x^20 + 305*x^16 - 8*x^15 + 7400*x^13 + 12310*x^12 + 880*x^11 - 351784*x^10 - 629200*x^9 - 46190*x^8 - 840*x^7 + 86410940*x^6 - 1433680*x^5 + 11025*x^4 + 5598132760*x^3 + 3006968580*x^2 + 224116200*x + 103753782505, 1)
 

Normalized defining polynomial

\( x^{20} + 305 x^{16} - 8 x^{15} + 7400 x^{13} + 12310 x^{12} + 880 x^{11} - 351784 x^{10} - 629200 x^{9} - 46190 x^{8} - 840 x^{7} + 86410940 x^{6} - 1433680 x^{5} + 11025 x^{4} + 5598132760 x^{3} + 3006968580 x^{2} + 224116200 x + 103753782505 \)

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:  \(10675123826048640000000000000000000000=2^{28}\cdot 3^{10}\cdot 5^{22}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.03$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{18} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{19} + \frac{518360102597541164341827170108931898685363112355964981283550806409880406840585492192139464836983}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{18} - \frac{393358991695716204023581584445956334408197985719382726474465430439485743123242660426127652983793}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{17} - \frac{1432836106229107209351688020708886551229865491533792984422290048493015688295863574683509212509767}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{16} - \frac{495852740041193003375903511859687383203640924714430778309640713273603899815768461911772329876263}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{15} + \frac{1580352601666305203665205049999848569125636025907042883190560977104562654156481376247684892083133}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{14} - \frac{3723806306537965582349737139014452751136048010416114272664364462425593855687530403951043772006467}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{13} - \frac{1066527742351031962232399816197282877821242005756732492203007973455644998463697621883361608690749}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{12} + \frac{3544788694112110796217752321743123276827588565801122560754969572561634877723430531790301993708719}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{11} - \frac{3908278407793156062089670076088776958705303768548650020898187168631336996324043071453285930772775}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{10} - \frac{2780164373896529910941550075270982715718738140118731677707915788640261757839523879030989787563349}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{9} + \frac{467442694726328812639791816756374006619077743779247432009098617213286076507156444728584344782775}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{8} + \frac{9828342324912774826680025688285995954708693196522696134135188809076611083496382777198530104798977}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{7} - \frac{6655596460864565988806601403308417056460627894306518880559499265460032267979301297480066792480115}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{6} + \frac{588823399059459989021152818586097151926909587268119864050041587452184100571905910545530887289715}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{5} - \frac{9639553762064524386133833621597415103839502622799593742770485489110674065905738620929425081164455}{21200852861117642992569062066762596755074708604609478449269393347440699366113563364697894326958844} a^{4} + \frac{3093445528642483140403814905425979358932405843814153993267854899556926582775485415879416422851153}{10600426430558821496284531033381298377537354302304739224634696673720349683056781682348947163479422} a^{3} + \frac{1652428754721407776588972913764690516344580283055312022267709370385905513517603223170755107126759}{10600426430558821496284531033381298377537354302304739224634696673720349683056781682348947163479422} a^{2} - \frac{4668112039024187135521993951277068853899327251695040824413961623904906645411118533768773816814563}{10600426430558821496284531033381298377537354302304739224634696673720349683056781682348947163479422} a - \frac{638023944938159894586078118365366696863115861559586435210209092367762749266803274241975800960028}{5300213215279410748142265516690649188768677151152369612317348336860174841528390841174473581739711}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-21})\), 5.1.50000.1, 10.0.3267280800000000000.1, 10.0.653456160000000000.1, 10.2.12500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
$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$$[\ ]_{2}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$