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Properties

Label 20.0.73445842291...4192.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{10}\cdot 11^{13}$
Root discriminant $55.37$
Ramified primes $2, 3, 11$
Class number Not computed
Class group Not computed
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![314911118323, 100071545420, 116894317946, 18926594408, 22516220277, 1894261020, 2845773570, 68785752, 277175770, -3538344, 21478084, -942756, 1344183, -78616, 66002, -4256, 2429, -144, 66, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 66*x^18 - 144*x^17 + 2429*x^16 - 4256*x^15 + 66002*x^14 - 78616*x^13 + 1344183*x^12 - 942756*x^11 + 21478084*x^10 - 3538344*x^9 + 277175770*x^8 + 68785752*x^7 + 2845773570*x^6 + 1894261020*x^5 + 22516220277*x^4 + 18926594408*x^3 + 116894317946*x^2 + 100071545420*x + 314911118323)
 
gp: K = bnfinit(x^20 - 4*x^19 + 66*x^18 - 144*x^17 + 2429*x^16 - 4256*x^15 + 66002*x^14 - 78616*x^13 + 1344183*x^12 - 942756*x^11 + 21478084*x^10 - 3538344*x^9 + 277175770*x^8 + 68785752*x^7 + 2845773570*x^6 + 1894261020*x^5 + 22516220277*x^4 + 18926594408*x^3 + 116894317946*x^2 + 100071545420*x + 314911118323, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 66 x^{18} - 144 x^{17} + 2429 x^{16} - 4256 x^{15} + 66002 x^{14} - 78616 x^{13} + 1344183 x^{12} - 942756 x^{11} + 21478084 x^{10} - 3538344 x^{9} + 277175770 x^{8} + 68785752 x^{7} + 2845773570 x^{6} + 1894261020 x^{5} + 22516220277 x^{4} + 18926594408 x^{3} + 116894317946 x^{2} + 100071545420 x + 314911118323 \)

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:  \(73445842291921443780364922039304192=2^{55}\cdot 3^{10}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.37$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} - \frac{4}{19} a^{16} - \frac{8}{19} a^{15} - \frac{2}{19} a^{14} + \frac{1}{19} a^{13} + \frac{7}{19} a^{12} + \frac{2}{19} a^{11} + \frac{2}{19} a^{10} + \frac{6}{19} a^{9} + \frac{7}{19} a^{8} - \frac{5}{19} a^{7} - \frac{8}{19} a^{6} + \frac{9}{19} a^{5} - \frac{6}{19} a^{4} + \frac{4}{19} a^{2} - \frac{7}{19} a + \frac{5}{19}$, $\frac{1}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{19} + \frac{1089625298067632601119467535516437301422369981382990319741913171366636723683107772711655072218836390}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{18} - \frac{176782124066845919591951065928871816273784561184741649346783824611389095443441426010008959163302954393}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{17} + \frac{107089022461625632246207419564885528808370504018375938490037976998624324009338031216308356855662428044}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{16} + \frac{294692546963120662798498015473498534298092198580139027435328708481886293376242127394859733350900473777}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{15} + \frac{125840026424918063291361606159907715747501422124130590548485905867048082266617040589668843599384319154}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{14} + \frac{256541632051460147878460226048629831315785921296797523865842943082102409885599154273080022211538633336}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{13} - \frac{533423943740206076367960767271443627854768072875270840830060584755693188278595498468480682604275548943}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{12} + \frac{444018518901045589849785568675434942571982167724614928386219196274550629936793907407290044702096637749}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{11} + \frac{283079576248411511923185510124676680927124065641639679886484967020838381470102760135220164675551550849}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{10} + \frac{381143489926628787373954274066435157559844267621162110006135254938153393175185239638219234962385040755}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{9} + \frac{74271814453702893616897602370614838428117637990217073098361999761211735900577999420069068387508465404}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{8} + \frac{496979007958296419399580571417654322928663571228288147959591419213580029868506621939170735192796811324}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{7} + \frac{292023182442961837808414982846354697313035543612949950887150213929333098128933392780743948039763556890}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{6} - \frac{153407995905425622410860705328145655067203702309106744214183365507803359962282232492274347736181559841}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{5} + \frac{849351436998731081327454515693995786850092912991319622935425527961887998460800176548070166830300566}{11198248839460042363246877754096163246707765359627262318772850092961770546952384409272579920957929547} a^{4} - \frac{69211353815962402915511015299053198405983049409627586689859069715383313866955896417266393549956085587}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{3} + \frac{360651343580846442016373639706186881219079982649597243431710731345092990114670578528785033442937471183}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a^{2} + \frac{110766016614033032512660150925933697370530766319221433118899687119573611165060011790325200941066138503}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059} a - \frac{67650066652612637993958873943461694647179089682822525419601284984341999367247834377641478385592369039}{1086230137427624109234947142147327834930653239883844444920966459017291743054381287699440252332919166059}$

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

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), 4.0.202752.6, 10.0.479756288.1

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 R R $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.8.3$x^{10} - 11 x^{5} + 847$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$