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Properties

Label 20.0.10080247818...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{17}\cdot 6029^{4}$
Root discriminant $22.40$
Ramified primes $5, 6029$
Class number $2$
Class group $[2]$
Galois group $C_4\times S_5$ (as 20T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 8, -67, 227, -128, 1006, -261, 1380, -217, 945, -358, 475, -301, 293, -177, 108, -42, 17, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 17*x^18 - 42*x^17 + 108*x^16 - 177*x^15 + 293*x^14 - 301*x^13 + 475*x^12 - 358*x^11 + 945*x^10 - 217*x^9 + 1380*x^8 - 261*x^7 + 1006*x^6 - 128*x^5 + 227*x^4 - 67*x^3 + 8*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 17*x^18 - 42*x^17 + 108*x^16 - 177*x^15 + 293*x^14 - 301*x^13 + 475*x^12 - 358*x^11 + 945*x^10 - 217*x^9 + 1380*x^8 - 261*x^7 + 1006*x^6 - 128*x^5 + 227*x^4 - 67*x^3 + 8*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 17 x^{18} - 42 x^{17} + 108 x^{16} - 177 x^{15} + 293 x^{14} - 301 x^{13} + 475 x^{12} - 358 x^{11} + 945 x^{10} - 217 x^{9} + 1380 x^{8} - 261 x^{7} + 1006 x^{6} - 128 x^{5} + 227 x^{4} - 67 x^{3} + 8 x^{2} + 6 x + 1 \)

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:  \(1008024781832337188720703125=5^{17}\cdot 6029^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.40$
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:  $5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{1}{5} a^{12} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{56062626132411290386294205} a^{19} - \frac{2157755387512456424189033}{56062626132411290386294205} a^{18} + \frac{4987824241110194969217306}{56062626132411290386294205} a^{17} - \frac{2386072664974141848749228}{56062626132411290386294205} a^{16} + \frac{1664638921008172630779226}{56062626132411290386294205} a^{15} - \frac{1095559602753155420812428}{11212525226482258077258841} a^{14} + \frac{7707083879458893131381532}{56062626132411290386294205} a^{13} + \frac{21365515963991383223129528}{56062626132411290386294205} a^{12} - \frac{2675144385957318318519734}{56062626132411290386294205} a^{11} + \frac{24927563556468506149734698}{56062626132411290386294205} a^{10} - \frac{4118078284344545241569236}{56062626132411290386294205} a^{9} - \frac{4114551583455930590530512}{56062626132411290386294205} a^{8} + \frac{4098511311583540180080204}{56062626132411290386294205} a^{7} - \frac{1338782233734336849013491}{11212525226482258077258841} a^{6} - \frac{7097081607590415494602343}{56062626132411290386294205} a^{5} + \frac{12683062282295017570969894}{56062626132411290386294205} a^{4} + \frac{1214897637055726083821243}{11212525226482258077258841} a^{3} - \frac{12791834501189988093895793}{56062626132411290386294205} a^{2} + \frac{3871476289413192930845225}{11212525226482258077258841} a + \frac{6987099983714744162367341}{56062626132411290386294205}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{7076469780059579555795616}{56062626132411290386294205} a^{19} - \frac{7393570948242118269789685}{11212525226482258077258841} a^{18} + \frac{128099567278025584365330253}{56062626132411290386294205} a^{17} - \frac{323763547733391614057864263}{56062626132411290386294205} a^{16} + \frac{829929107853981644173077533}{56062626132411290386294205} a^{15} - \frac{1422767867673938658915564979}{56062626132411290386294205} a^{14} + \frac{2351244544815771343632815599}{56062626132411290386294205} a^{13} - \frac{2599969058902110447195651486}{56062626132411290386294205} a^{12} + \frac{3848940476657224104432984928}{56062626132411290386294205} a^{11} - \frac{3326300217118276224909281277}{56062626132411290386294205} a^{10} + \frac{7277016397608970193798801407}{56062626132411290386294205} a^{9} - \frac{621760971892204484744519837}{11212525226482258077258841} a^{8} + \frac{10085952520796098846874643376}{56062626132411290386294205} a^{7} - \frac{4238684661711000131611176586}{56062626132411290386294205} a^{6} + \frac{7439697549052160064550078454}{56062626132411290386294205} a^{5} - \frac{2792381264250064278071500706}{56062626132411290386294205} a^{4} + \frac{1773465667294405773318308414}{56062626132411290386294205} a^{3} - \frac{1100936551529003795661285187}{56062626132411290386294205} a^{2} + \frac{162850912008780705965746269}{56062626132411290386294205} a + \frac{8617190728072315981589776}{56062626132411290386294205} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 341439.528105 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times S_5$ (as 20T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 28 conjugacy class representatives for $C_4\times S_5$
Character table for $C_4\times S_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.753625.1, 10.10.2839753203125.1

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 24 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 $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
6029Data not computed