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Properties

Label 20.4.92301252341...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{15}\cdot 19^{16}$
Root discriminant $70.51$
Ramified primes $2, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50000, 0, 67500, 0, 733025, 0, -1005950, 0, 127585, 0, -36365, 0, -2014, 0, 149, 0, -109, 0, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^18 - 109*x^16 + 149*x^14 - 2014*x^12 - 36365*x^10 + 127585*x^8 - 1005950*x^6 + 733025*x^4 + 67500*x^2 + 50000)
 
gp: K = bnfinit(x^20 - x^18 - 109*x^16 + 149*x^14 - 2014*x^12 - 36365*x^10 + 127585*x^8 - 1005950*x^6 + 733025*x^4 + 67500*x^2 + 50000, 1)
 

Normalized defining polynomial

\( x^{20} - x^{18} - 109 x^{16} + 149 x^{14} - 2014 x^{12} - 36365 x^{10} + 127585 x^{8} - 1005950 x^{6} + 733025 x^{4} + 67500 x^{2} + 50000 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9230125234163877365792000000000000000=2^{20}\cdot 5^{15}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.51$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4}$, $\frac{1}{10} a^{13} + \frac{2}{5} a^{11} - \frac{2}{5} a^{9} - \frac{1}{10} a^{7} + \frac{1}{10} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{14} - \frac{1}{2} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{9}{20} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{100} a^{15} + \frac{1}{25} a^{13} + \frac{3}{50} a^{11} - \frac{41}{100} a^{9} - \frac{49}{100} a^{7} - \frac{1}{20} a^{5} + \frac{1}{20} a^{3}$, $\frac{1}{5700} a^{16} - \frac{23}{2850} a^{14} - \frac{187}{2850} a^{12} + \frac{613}{1900} a^{10} + \frac{1921}{5700} a^{8} + \frac{25}{228} a^{6} + \frac{1}{76} a^{4} + \frac{15}{38} a^{2} - \frac{22}{57}$, $\frac{1}{5700} a^{17} + \frac{11}{5700} a^{15} - \frac{73}{2850} a^{13} + \frac{727}{1900} a^{11} - \frac{104}{1425} a^{9} - \frac{542}{1425} a^{7} - \frac{7}{190} a^{5} + \frac{169}{380} a^{3} - \frac{22}{57} a$, $\frac{1}{13137516879319917006913500} a^{18} - \frac{109255244309240490269}{3284379219829979251728375} a^{16} + \frac{677603995817475657479}{3284379219829979251728375} a^{14} - \frac{1153462523441771570749201}{13137516879319917006913500} a^{12} + \frac{318500976585395017018711}{13137516879319917006913500} a^{10} + \frac{232059015715361672408719}{875834458621327800460900} a^{8} + \frac{526599275775527417053747}{2627503375863983401382700} a^{6} - \frac{26187906337942557076301}{87583445862132780046090} a^{4} - \frac{40627632354161818057367}{262750337586398340138270} a^{2} + \frac{6635001177105407686682}{26275033758639834013827}$, $\frac{1}{131375168793199170069135000} a^{19} + \frac{11087116636201561729199}{131375168793199170069135000} a^{17} + \frac{129475929731093663222941}{131375168793199170069135000} a^{15} - \frac{69494413046604209382217}{43791722931066390023045000} a^{13} - \frac{8293704451164459618307357}{65687584396599585034567500} a^{11} + \frac{8145379600472746730619917}{26275033758639834013827000} a^{9} + \frac{2713916470245234623862489}{8758344586213278004609000} a^{7} + \frac{37590470474983906464678}{218958614655331950115225} a^{5} - \frac{2591212436915234095856629}{5255006751727966802765400} a^{3} - \frac{4689749205377595370957}{17516689172426556009218} a$

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:  $11$
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:  \( 218846902353.9545 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 5.1.16290125.1, 10.2.1326840862578125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$19$19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$