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Properties

Label 20.8.19481352064...8125.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{8}\cdot 5^{11}\cdot 239^{10}$
Root discriminant $58.14$
Ramified primes $3, 5, 239$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1251, 5679, 4401, -18411, -20012, 23307, 46116, -7150, -131112, 177933, -88217, -9523, 32014, -16503, 3029, 949, -606, 126, -8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 8*x^18 + 126*x^17 - 606*x^16 + 949*x^15 + 3029*x^14 - 16503*x^13 + 32014*x^12 - 9523*x^11 - 88217*x^10 + 177933*x^9 - 131112*x^8 - 7150*x^7 + 46116*x^6 + 23307*x^5 - 20012*x^4 - 18411*x^3 + 4401*x^2 + 5679*x + 1251)
 
gp: K = bnfinit(x^20 - 5*x^19 - 8*x^18 + 126*x^17 - 606*x^16 + 949*x^15 + 3029*x^14 - 16503*x^13 + 32014*x^12 - 9523*x^11 - 88217*x^10 + 177933*x^9 - 131112*x^8 - 7150*x^7 + 46116*x^6 + 23307*x^5 - 20012*x^4 - 18411*x^3 + 4401*x^2 + 5679*x + 1251, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 8 x^{18} + 126 x^{17} - 606 x^{16} + 949 x^{15} + 3029 x^{14} - 16503 x^{13} + 32014 x^{12} - 9523 x^{11} - 88217 x^{10} + 177933 x^{9} - 131112 x^{8} - 7150 x^{7} + 46116 x^{6} + 23307 x^{5} - 20012 x^{4} - 18411 x^{3} + 4401 x^{2} + 5679 x + 1251 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(194813520645626483248608015673828125=3^{8}\cdot 5^{11}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.14$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{411} a^{16} - \frac{4}{411} a^{15} + \frac{43}{411} a^{14} + \frac{21}{137} a^{13} + \frac{10}{411} a^{12} - \frac{61}{137} a^{11} + \frac{83}{411} a^{10} + \frac{20}{137} a^{9} + \frac{25}{411} a^{8} + \frac{154}{411} a^{7} + \frac{89}{411} a^{6} - \frac{37}{137} a^{5} + \frac{82}{411} a^{4} - \frac{62}{137} a^{3} + \frac{124}{411} a^{2} + \frac{57}{137} a - \frac{20}{137}$, $\frac{1}{411} a^{17} + \frac{9}{137} a^{15} - \frac{13}{137} a^{14} - \frac{4}{137} a^{13} - \frac{2}{137} a^{12} + \frac{12}{137} a^{11} - \frac{19}{411} a^{10} - \frac{146}{411} a^{9} - \frac{20}{411} a^{8} - \frac{39}{137} a^{7} - \frac{166}{411} a^{6} - \frac{88}{411} a^{5} + \frac{142}{411} a^{4} - \frac{24}{137} a^{3} - \frac{6}{137} a^{2} - \frac{66}{137} a + \frac{57}{137}$, $\frac{1}{1598379} a^{18} + \frac{482}{532793} a^{17} - \frac{1607}{1598379} a^{16} - \frac{118313}{1598379} a^{15} + \frac{25501}{532793} a^{14} + \frac{50183}{532793} a^{13} + \frac{45012}{532793} a^{12} - \frac{652192}{1598379} a^{11} - \frac{355316}{1598379} a^{10} + \frac{144957}{532793} a^{9} + \frac{81007}{532793} a^{8} + \frac{114001}{1598379} a^{7} - \frac{118470}{532793} a^{6} - \frac{147498}{532793} a^{5} - \frac{519746}{1598379} a^{4} - \frac{195894}{532793} a^{3} - \frac{352553}{1598379} a^{2} + \frac{2050}{532793} a - \frac{21076}{532793}$, $\frac{1}{21262086953517917366657174322066138342447} a^{19} - \frac{2212182278835729848077716847990248}{7087362317839305788885724774022046114149} a^{18} + \frac{21869867435476056148841188182510612265}{21262086953517917366657174322066138342447} a^{17} + \frac{6085971450840413729732842525738475282}{21262086953517917366657174322066138342447} a^{16} + \frac{2456617535146474272536565646077207119434}{21262086953517917366657174322066138342447} a^{15} + \frac{990360323734360325528866817694357716911}{21262086953517917366657174322066138342447} a^{14} - \frac{701109165319449752310103182123794753866}{21262086953517917366657174322066138342447} a^{13} + \frac{2554984968600962234035461567826987930109}{21262086953517917366657174322066138342447} a^{12} - \frac{4131936161678789063284995632205261654560}{21262086953517917366657174322066138342447} a^{11} + \frac{357773543473802708037897465553549125145}{21262086953517917366657174322066138342447} a^{10} + \frac{3095105024580010578718553221231524610649}{7087362317839305788885724774022046114149} a^{9} + \frac{9281666097828839287931640762922452747799}{21262086953517917366657174322066138342447} a^{8} - \frac{1897056127990519158814243397725132732354}{21262086953517917366657174322066138342447} a^{7} + \frac{1654966942964404473528753167424708854033}{7087362317839305788885724774022046114149} a^{6} + \frac{2459366109829312540777547218843337863421}{21262086953517917366657174322066138342447} a^{5} + \frac{7726566397256679621404495014209055860295}{21262086953517917366657174322066138342447} a^{4} - \frac{7543965872793516328460593659502825591001}{21262086953517917366657174322066138342447} a^{3} + \frac{2775843269174193414421221517988877647137}{7087362317839305788885724774022046114149} a^{2} + \frac{775221281974993123483432307973051858758}{7087362317839305788885724774022046114149} a - \frac{2223781615492305266658260797962271903756}{7087362317839305788885724774022046114149}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $13$
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:  \( 4395614780.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed