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Properties

Label 20.4.79637947727...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{4}\cdot 5^{11}\cdot 29^{5}\cdot 89^{6}$
Root discriminant $24.83$
Ramified primes $2, 5, 29, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, 80, -120, 56, 307, -1003, 39, 1056, 155, -92, -924, -51, 698, -67, -174, -9, 41, 5, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 5*x^18 + 5*x^17 + 41*x^16 - 9*x^15 - 174*x^14 - 67*x^13 + 698*x^12 - 51*x^11 - 924*x^10 - 92*x^9 + 155*x^8 + 1056*x^7 + 39*x^6 - 1003*x^5 + 307*x^4 + 56*x^3 - 120*x^2 + 80*x - 16)
 
gp: K = bnfinit(x^20 - 2*x^19 - 5*x^18 + 5*x^17 + 41*x^16 - 9*x^15 - 174*x^14 - 67*x^13 + 698*x^12 - 51*x^11 - 924*x^10 - 92*x^9 + 155*x^8 + 1056*x^7 + 39*x^6 - 1003*x^5 + 307*x^4 + 56*x^3 - 120*x^2 + 80*x - 16, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 5 x^{18} + 5 x^{17} + 41 x^{16} - 9 x^{15} - 174 x^{14} - 67 x^{13} + 698 x^{12} - 51 x^{11} - 924 x^{10} - 92 x^{9} + 155 x^{8} + 1056 x^{7} + 39 x^{6} - 1003 x^{5} + 307 x^{4} + 56 x^{3} - 120 x^{2} + 80 x - 16 \)

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:  \(7963794772744862647656250000=2^{4}\cdot 5^{11}\cdot 29^{5}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.83$
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:  $2, 5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} + \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{80} a^{18} - \frac{1}{8} a^{17} - \frac{1}{80} a^{16} - \frac{27}{80} a^{15} + \frac{13}{80} a^{14} + \frac{19}{80} a^{13} - \frac{17}{40} a^{12} - \frac{39}{80} a^{11} - \frac{3}{40} a^{10} + \frac{1}{80} a^{9} + \frac{1}{20} a^{8} - \frac{1}{2} a^{7} + \frac{11}{80} a^{6} + \frac{1}{10} a^{5} + \frac{19}{80} a^{4} - \frac{3}{80} a^{3} + \frac{7}{80} a^{2} - \frac{3}{20} a + \frac{1}{20}$, $\frac{1}{18950415339219275652322240} a^{19} + \frac{3823183819076523970735}{947520766960963782616112} a^{18} + \frac{811856459706564438364099}{18950415339219275652322240} a^{17} + \frac{3638593036678797486555343}{18950415339219275652322240} a^{16} + \frac{5039318316092732067958203}{18950415339219275652322240} a^{15} + \frac{7114559898088458428803569}{18950415339219275652322240} a^{14} + \frac{88204622341884045454224}{296100239675301182067535} a^{13} + \frac{7123101700801677668089181}{18950415339219275652322240} a^{12} + \frac{561967392555141514608079}{1184400958701204728270140} a^{11} - \frac{6998545918020318485896819}{18950415339219275652322240} a^{10} - \frac{3056045392475741329001123}{9475207669609637826161120} a^{9} - \frac{67845105136323534904985}{236880191740240945654028} a^{8} + \frac{6515841161578411333678651}{18950415339219275652322240} a^{7} - \frac{3795433754638630854977731}{9475207669609637826161120} a^{6} + \frac{9367405603692518247665459}{18950415339219275652322240} a^{5} - \frac{1693275001856962223900353}{18950415339219275652322240} a^{4} + \frac{7985080013686843437902117}{18950415339219275652322240} a^{3} + \frac{3666120571112200192563999}{9475207669609637826161120} a^{2} + \frac{925092365865204457275091}{4737603834804818913080560} a - \frac{194080584012649529336341}{473760383480481891308056}$

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:  \( 2007537.24297 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.1852746653125.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R $16{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ R ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.4.3.1$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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.8.0.1$x^{8} - x + 62$$1$$8$$0$$C_8$$[\ ]^{8}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$