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Properties

Label 20.20.8308610292...3888.2
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 3^{10}\cdot 7^{8}\cdot 11^{8}\cdot 13^{9}$
Root discriminant $88.30$
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![-1367, -13042, 87522, 81360, -1003929, 1263226, 1031692, -2699058, 475396, 1688544, -789498, -384760, 265203, 32222, -36852, -800, 2529, -8, -84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 84*x^18 - 8*x^17 + 2529*x^16 - 800*x^15 - 36852*x^14 + 32222*x^13 + 265203*x^12 - 384760*x^11 - 789498*x^10 + 1688544*x^9 + 475396*x^8 - 2699058*x^7 + 1031692*x^6 + 1263226*x^5 - 1003929*x^4 + 81360*x^3 + 87522*x^2 - 13042*x - 1367)
 
gp: K = bnfinit(x^20 - 84*x^18 - 8*x^17 + 2529*x^16 - 800*x^15 - 36852*x^14 + 32222*x^13 + 265203*x^12 - 384760*x^11 - 789498*x^10 + 1688544*x^9 + 475396*x^8 - 2699058*x^7 + 1031692*x^6 + 1263226*x^5 - 1003929*x^4 + 81360*x^3 + 87522*x^2 - 13042*x - 1367, 1)
 

Normalized defining polynomial

\( x^{20} - 84 x^{18} - 8 x^{17} + 2529 x^{16} - 800 x^{15} - 36852 x^{14} + 32222 x^{13} + 265203 x^{12} - 384760 x^{11} - 789498 x^{10} + 1688544 x^{9} + 475396 x^{8} - 2699058 x^{7} + 1031692 x^{6} + 1263226 x^{5} - 1003929 x^{4} + 81360 x^{3} + 87522 x^{2} - 13042 x - 1367 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(830861029216363101082988400989215653888=2^{30}\cdot 3^{10}\cdot 7^{8}\cdot 11^{8}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.30$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} + \frac{4}{13} a^{15} - \frac{4}{13} a^{14} + \frac{3}{13} a^{12} - \frac{2}{13} a^{11} + \frac{4}{13} a^{10} + \frac{6}{13} a^{9} - \frac{3}{13} a^{8} - \frac{3}{13} a^{7} - \frac{4}{13} a^{6} + \frac{2}{13} a^{5} + \frac{1}{13} a^{4} + \frac{5}{13} a^{3} - \frac{5}{13} a^{2} - \frac{5}{13}$, $\frac{1}{13} a^{17} + \frac{6}{13} a^{15} + \frac{3}{13} a^{14} + \frac{3}{13} a^{13} - \frac{1}{13} a^{12} - \frac{1}{13} a^{11} + \frac{3}{13} a^{10} - \frac{1}{13} a^{9} - \frac{4}{13} a^{8} - \frac{5}{13} a^{7} + \frac{5}{13} a^{6} + \frac{6}{13} a^{5} + \frac{1}{13} a^{4} + \frac{1}{13} a^{3} - \frac{6}{13} a^{2} - \frac{5}{13} a - \frac{6}{13}$, $\frac{1}{169} a^{18} - \frac{3}{169} a^{17} + \frac{4}{169} a^{16} - \frac{36}{169} a^{15} + \frac{28}{169} a^{14} - \frac{75}{169} a^{13} - \frac{82}{169} a^{12} - \frac{3}{169} a^{11} - \frac{44}{169} a^{10} - \frac{6}{13} a^{8} - \frac{5}{13} a^{7} + \frac{64}{169} a^{6} + \frac{18}{169} a^{5} + \frac{9}{169} a^{4} + \frac{33}{169} a^{3} + \frac{36}{169} a^{2} - \frac{4}{169} a + \frac{80}{169}$, $\frac{1}{47969149565113489924971685721379742554258031} a^{19} - \frac{48649914797827254108426130735264239468642}{47969149565113489924971685721379742554258031} a^{18} - \frac{846618814436257282994351887346070261807373}{47969149565113489924971685721379742554258031} a^{17} + \frac{890284537331572492311500342881769991634507}{47969149565113489924971685721379742554258031} a^{16} - \frac{1650327768899642012006734651341313023278744}{3689934581931806917305514286259980196481387} a^{15} + \frac{2119234694370910804687091682570180921756548}{47969149565113489924971685721379742554258031} a^{14} - \frac{3091000535734008686140146181226141171497140}{47969149565113489924971685721379742554258031} a^{13} + \frac{8326949427365077110948926069896426553894014}{47969149565113489924971685721379742554258031} a^{12} - \frac{17070546808241526131063795269339489893883682}{47969149565113489924971685721379742554258031} a^{11} + \frac{8364904132745346252408976192233370227134043}{47969149565113489924971685721379742554258031} a^{10} - \frac{1091302721468811448269981109304756644572257}{3689934581931806917305514286259980196481387} a^{9} - \frac{1045201402249978685581413699871949745126249}{3689934581931806917305514286259980196481387} a^{8} + \frac{19252233785243950053464625993573958751398404}{47969149565113489924971685721379742554258031} a^{7} - \frac{12008346466619791933023783082935266877965114}{47969149565113489924971685721379742554258031} a^{6} - \frac{18417959063574499904412269113116105831042284}{47969149565113489924971685721379742554258031} a^{5} - \frac{13040311244611611981481100367447891555828963}{47969149565113489924971685721379742554258031} a^{4} + \frac{10639031559343668334625361793863566037139568}{47969149565113489924971685721379742554258031} a^{3} - \frac{941400271162041944939406110806762379603897}{47969149565113489924971685721379742554258031} a^{2} - \frac{19276347525005569482255007048786332047112051}{47969149565113489924971685721379742554258031} a + \frac{5645490846315568642083085491234778242007641}{47969149565113489924971685721379742554258031}$

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:  $19$
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:  \( 18356338138900 \) (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

\(\Q(\sqrt{3}) \), 4.4.7488.1, 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 ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{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}$