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Properties

Label 20.8.21061833705...4848.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{10}\cdot 19^{12}\cdot 43^{11}$
Root discriminant $65.49$
Ramified primes $2, 19, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-953, 11642, -1163, 28599, -148930, -34374, 315447, -83236, -122359, -30992, -15946, 108760, -48227, 3573, -1548, 814, 197, -81, 5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 5*x^18 - 81*x^17 + 197*x^16 + 814*x^15 - 1548*x^14 + 3573*x^13 - 48227*x^12 + 108760*x^11 - 15946*x^10 - 30992*x^9 - 122359*x^8 - 83236*x^7 + 315447*x^6 - 34374*x^5 - 148930*x^4 + 28599*x^3 - 1163*x^2 + 11642*x - 953)
 
gp: K = bnfinit(x^20 - 4*x^19 + 5*x^18 - 81*x^17 + 197*x^16 + 814*x^15 - 1548*x^14 + 3573*x^13 - 48227*x^12 + 108760*x^11 - 15946*x^10 - 30992*x^9 - 122359*x^8 - 83236*x^7 + 315447*x^6 - 34374*x^5 - 148930*x^4 + 28599*x^3 - 1163*x^2 + 11642*x - 953, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 5 x^{18} - 81 x^{17} + 197 x^{16} + 814 x^{15} - 1548 x^{14} + 3573 x^{13} - 48227 x^{12} + 108760 x^{11} - 15946 x^{10} - 30992 x^{9} - 122359 x^{8} - 83236 x^{7} + 315447 x^{6} - 34374 x^{5} - 148930 x^{4} + 28599 x^{3} - 1163 x^{2} + 11642 x - 953 \)

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:  \(2106183370512833937071945009682254848=2^{10}\cdot 19^{12}\cdot 43^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.49$
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:  $2, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{15} a^{17} + \frac{1}{15} a^{16} - \frac{1}{15} a^{15} - \frac{1}{3} a^{13} + \frac{2}{15} a^{12} + \frac{7}{15} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{8} + \frac{1}{15} a^{7} + \frac{1}{15} a^{5} + \frac{1}{15} a^{4} - \frac{4}{15} a^{3} + \frac{2}{15} a^{2} + \frac{1}{15} a - \frac{1}{15}$, $\frac{1}{645} a^{18} - \frac{13}{645} a^{17} + \frac{1}{129} a^{16} + \frac{19}{645} a^{15} - \frac{11}{215} a^{13} + \frac{13}{215} a^{12} - \frac{227}{645} a^{11} + \frac{12}{215} a^{10} + \frac{233}{645} a^{9} - \frac{196}{645} a^{8} + \frac{271}{645} a^{7} - \frac{214}{645} a^{6} - \frac{76}{215} a^{5} + \frac{14}{215} a^{4} + \frac{223}{645} a^{3} + \frac{51}{215} a^{2} + \frac{37}{129} a - \frac{87}{215}$, $\frac{1}{107661143308133792566574373378556057521625834026678945} a^{19} + \frac{23263782265107754702431228862126877958270861907714}{35887047769377930855524791126185352507208611342226315} a^{18} - \frac{214254824519943880977941319866688561423506971982986}{8281626408317984043582644106042773655509679540513765} a^{17} - \frac{5339676802792467748064190455507557298516991888104044}{107661143308133792566574373378556057521625834026678945} a^{16} - \frac{261987421066711712704486396705411300865542073370882}{5126721109911132979360684446597907501029801620318045} a^{15} - \frac{39788865644975687221629491757115262554556410984236158}{107661143308133792566574373378556057521625834026678945} a^{14} - \frac{5342477846565608075712610206188759908654008496962082}{35887047769377930855524791126185352507208611342226315} a^{13} - \frac{743159035166604129687976785839835405478582235839646}{2503747518793809129455217985547815291200600791318115} a^{12} + \frac{2303522312193017639676828989180022364448585858915}{21532228661626758513314874675711211504325166805335789} a^{11} - \frac{1955383815970875464385171569183889554114831381047835}{7177409553875586171104958225237070501441722268445263} a^{10} - \frac{42774622193050888364613220519420549925498445564821231}{107661143308133792566574373378556057521625834026678945} a^{9} + \frac{37839135520784664713935571883019937809083169345052102}{107661143308133792566574373378556057521625834026678945} a^{8} + \frac{27488478179897217906834787580906973080087304245894348}{107661143308133792566574373378556057521625834026678945} a^{7} - \frac{11032423862762173077532980899770196319611967284220066}{35887047769377930855524791126185352507208611342226315} a^{6} - \frac{47257677550786663210414044727774354981786076028968926}{107661143308133792566574373378556057521625834026678945} a^{5} + \frac{563217687472423152890340960211009247238774837722216}{3076032665946679787616410667958744500617880972190827} a^{4} + \frac{5269654725729458104829994199235491164819108164318}{38519192596827832760849507469966389095393858327971} a^{3} + \frac{155867715963054343735259757861548519978747613744964}{5126721109911132979360684446597907501029801620318045} a^{2} + \frac{36484592604044019910824512650140127350312896457460616}{107661143308133792566574373378556057521625834026678945} a + \frac{18669026250437488441367089474544981861787785027351513}{107661143308133792566574373378556057521625834026678945}$

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:  $13$
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:  \( 48319668355.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 is not computed

Intermediate fields

\(\Q(\sqrt{817}) \), 5.5.667489.1 x5, 10.10.364007458703857.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.10$x^{10} - 11 x^{8} + 10 x^{6} - 62 x^{4} + 21 x^{2} - 55$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$