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Properties

Label 20.10.8925264962...0032.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 3^{10}\cdot 7^{8}\cdot 11^{9}\cdot 13^{9}$
Root discriminant $70.39$
Ramified primes $2, 3, 7, 11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{10}^2:C_2^2$ (as 20T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![69001, -635210, 1541335, -523652, -2361935, 2289288, 764544, -1466670, 9741, 238244, 79121, 57640, -15111, -12260, 165, 150, 227, 82, -28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 28*x^18 + 82*x^17 + 227*x^16 + 150*x^15 + 165*x^14 - 12260*x^13 - 15111*x^12 + 57640*x^11 + 79121*x^10 + 238244*x^9 + 9741*x^8 - 1466670*x^7 + 764544*x^6 + 2289288*x^5 - 2361935*x^4 - 523652*x^3 + 1541335*x^2 - 635210*x + 69001)
 
gp: K = bnfinit(x^20 - 4*x^19 - 28*x^18 + 82*x^17 + 227*x^16 + 150*x^15 + 165*x^14 - 12260*x^13 - 15111*x^12 + 57640*x^11 + 79121*x^10 + 238244*x^9 + 9741*x^8 - 1466670*x^7 + 764544*x^6 + 2289288*x^5 - 2361935*x^4 - 523652*x^3 + 1541335*x^2 - 635210*x + 69001, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 28 x^{18} + 82 x^{17} + 227 x^{16} + 150 x^{15} + 165 x^{14} - 12260 x^{13} - 15111 x^{12} + 57640 x^{11} + 79121 x^{10} + 238244 x^{9} + 9741 x^{8} - 1466670 x^{7} + 764544 x^{6} + 2289288 x^{5} - 2361935 x^{4} - 523652 x^{3} + 1541335 x^{2} - 635210 x + 69001 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8925264962285150499914914463751340032=-\,2^{20}\cdot 3^{10}\cdot 7^{8}\cdot 11^{9}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.39$
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:  $2, 3, 7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{3}{11} a^{14} - \frac{5}{11} a^{13} - \frac{4}{11} a^{12} + \frac{5}{11} a^{10} - \frac{4}{11} a^{9} - \frac{5}{11} a^{8} - \frac{4}{11} a^{7} - \frac{5}{11} a^{6} - \frac{2}{11} a^{5} - \frac{4}{11} a^{4} + \frac{5}{11} a^{3} - \frac{1}{11} a^{2} - \frac{1}{11} a + \frac{2}{11}$, $\frac{1}{77} a^{16} + \frac{3}{77} a^{15} - \frac{12}{77} a^{14} + \frac{32}{77} a^{13} - \frac{2}{77} a^{12} + \frac{5}{77} a^{11} - \frac{18}{77} a^{10} - \frac{29}{77} a^{9} - \frac{23}{77} a^{8} + \frac{4}{77} a^{7} - \frac{10}{77} a^{6} - \frac{5}{77} a^{5} - \frac{19}{77} a^{4} + \frac{29}{77} a^{3} - \frac{1}{11} a^{2} - \frac{15}{77} a + \frac{1}{77}$, $\frac{1}{77} a^{17} + \frac{5}{77} a^{14} + \frac{4}{11} a^{13} + \frac{4}{77} a^{12} - \frac{3}{7} a^{11} - \frac{24}{77} a^{10} - \frac{20}{77} a^{9} - \frac{32}{77} a^{8} - \frac{29}{77} a^{7} - \frac{3}{77} a^{6} + \frac{31}{77} a^{5} + \frac{2}{77} a^{4} + \frac{1}{7} a^{3} - \frac{15}{77} a^{2} + \frac{25}{77} a - \frac{38}{77}$, $\frac{1}{1001} a^{18} + \frac{5}{1001} a^{17} - \frac{6}{1001} a^{16} + \frac{43}{1001} a^{15} - \frac{197}{1001} a^{14} + \frac{365}{1001} a^{13} - \frac{456}{1001} a^{12} + \frac{474}{1001} a^{11} - \frac{214}{1001} a^{10} - \frac{37}{143} a^{9} - \frac{331}{1001} a^{8} - \frac{36}{91} a^{7} + \frac{335}{1001} a^{6} + \frac{75}{1001} a^{5} - \frac{243}{1001} a^{4} + \frac{223}{1001} a^{3} + \frac{1}{77} a^{2} - \frac{10}{91} a + \frac{54}{143}$, $\frac{1}{3527212249949582717483878432779113576671423724043918119} a^{19} + \frac{153559883640466737246432027598281988718155907399135}{320655659086325701589443493889010325151947611276719829} a^{18} + \frac{7497750189600356976788467374228901454200198157021693}{3527212249949582717483878432779113576671423724043918119} a^{17} + \frac{308234881695094789581672630921834643273017855766221}{153357054345634031194951236207787546811801031480170353} a^{16} + \frac{146787411826578861721993550933397150287373750107919368}{3527212249949582717483878432779113576671423724043918119} a^{15} - \frac{161509029962243342861085353866206105295093748617586840}{3527212249949582717483878432779113576671423724043918119} a^{14} + \frac{78863942025866188259407581395934829635156070683334186}{271324019226890978267990648675316428974724901849532163} a^{13} + \frac{153487376414370179018716070434796341102830737381057345}{3527212249949582717483878432779113576671423724043918119} a^{12} - \frac{45591498056552265968570031649764261235272778446353049}{271324019226890978267990648675316428974724901849532163} a^{11} + \frac{805717570310045699935873414079237358711127690911301395}{3527212249949582717483878432779113576671423724043918119} a^{10} + \frac{1635526658231816095446839356994712709742119560763934121}{3527212249949582717483878432779113576671423724043918119} a^{9} + \frac{433909379153576720632177747747759640508746302625865017}{3527212249949582717483878432779113576671423724043918119} a^{8} - \frac{997803413725480157936687942381398025594418762091854524}{3527212249949582717483878432779113576671423724043918119} a^{7} + \frac{91155229185762353773850864146212873553616817335304638}{503887464278511816783411204682730510953060532006274017} a^{6} - \frac{662557089428386859015047803945021012899862872514648573}{3527212249949582717483878432779113576671423724043918119} a^{5} + \frac{13187422605782051088075015479365693390986838513353385}{3527212249949582717483878432779113576671423724043918119} a^{4} - \frac{1471665233156493021394494565673830513530004985649407937}{3527212249949582717483878432779113576671423724043918119} a^{3} - \frac{18367885748919053000009477220840645603047455902557813}{503887464278511816783411204682730510953060532006274017} a^{2} + \frac{524340622062311005583768156522860336484194715693114417}{3527212249949582717483878432779113576671423724043918119} a - \frac{799282312745891599119182037495213765645012530886991700}{3527212249949582717483878432779113576671423724043918119}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 156231018145 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}^2:C_2^2$ (as 20T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 46 conjugacy class representatives for $C_{10}^2:C_2^2$
Character table for $C_{10}^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.2.20592.4, 10.10.249828821987576832.1

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 $20$ R R R $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
3Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.9.3$x^{10} - 891$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$