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Properties

Label 20.12.4443128453...9392.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{16}\cdot 17^{3}\cdot 53^{14}$
Root discriminant $42.89$
Ramified primes $2, 17, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T513

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, 32, 408, -328, -2937, -300, 3832, 5232, 4480, -16280, -5674, 15684, -727, -4800, 1406, 28, -70, 52, -10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 10*x^18 + 52*x^17 - 70*x^16 + 28*x^15 + 1406*x^14 - 4800*x^13 - 727*x^12 + 15684*x^11 - 5674*x^10 - 16280*x^9 + 4480*x^8 + 5232*x^7 + 3832*x^6 - 300*x^5 - 2937*x^4 - 328*x^3 + 408*x^2 + 32*x - 16)
 
gp: K = bnfinit(x^20 - 4*x^19 - 10*x^18 + 52*x^17 - 70*x^16 + 28*x^15 + 1406*x^14 - 4800*x^13 - 727*x^12 + 15684*x^11 - 5674*x^10 - 16280*x^9 + 4480*x^8 + 5232*x^7 + 3832*x^6 - 300*x^5 - 2937*x^4 - 328*x^3 + 408*x^2 + 32*x - 16, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 10 x^{18} + 52 x^{17} - 70 x^{16} + 28 x^{15} + 1406 x^{14} - 4800 x^{13} - 727 x^{12} + 15684 x^{11} - 5674 x^{10} - 16280 x^{9} + 4480 x^{8} + 5232 x^{7} + 3832 x^{6} - 300 x^{5} - 2937 x^{4} - 328 x^{3} + 408 x^{2} + 32 x - 16 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(444312845384859473286477479739392=2^{16}\cdot 17^{3}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.89$
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:  $2, 17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} + \frac{3}{16} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{3}{16} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{48} a^{16} - \frac{1}{48} a^{14} + \frac{1}{24} a^{13} + \frac{1}{24} a^{12} - \frac{1}{24} a^{11} - \frac{1}{16} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{5}{24} a^{7} - \frac{1}{16} a^{6} - \frac{1}{24} a^{5} + \frac{1}{48} a^{4} + \frac{7}{24} a^{3} - \frac{5}{24} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{240} a^{17} - \frac{1}{60} a^{15} - \frac{1}{240} a^{14} - \frac{13}{240} a^{13} + \frac{13}{240} a^{12} - \frac{1}{80} a^{11} - \frac{3}{40} a^{10} - \frac{11}{80} a^{9} + \frac{5}{48} a^{8} + \frac{3}{80} a^{7} - \frac{19}{120} a^{6} - \frac{19}{120} a^{5} + \frac{29}{240} a^{4} - \frac{19}{240} a^{3} + \frac{59}{240} a^{2} + \frac{13}{60} a + \frac{1}{20}$, $\frac{1}{720} a^{18} + \frac{1}{720} a^{16} + \frac{7}{360} a^{15} - \frac{1}{40} a^{14} + \frac{1}{90} a^{13} + \frac{37}{720} a^{12} - \frac{11}{90} a^{11} + \frac{1}{60} a^{10} - \frac{1}{36} a^{9} - \frac{17}{240} a^{8} + \frac{1}{10} a^{7} - \frac{173}{720} a^{6} - \frac{7}{90} a^{5} + \frac{49}{180} a^{4} + \frac{3}{40} a^{3} + \frac{91}{360} a^{2} - \frac{8}{45} a + \frac{5}{18}$, $\frac{1}{8526632732369682320160} a^{19} + \frac{2349640979077952909}{4263316366184841160080} a^{18} + \frac{1715626994321456119}{2131658183092420580040} a^{17} - \frac{10495643992065187393}{1421105455394947053360} a^{16} + \frac{82868880005252960917}{4263316366184841160080} a^{15} + \frac{11340507334249271069}{532914545773105145010} a^{14} - \frac{13737691379446577249}{236850909232491175560} a^{13} + \frac{52952879944818529993}{1421105455394947053360} a^{12} + \frac{339580516988945142023}{8526632732369682320160} a^{11} - \frac{63545433360007626101}{2131658183092420580040} a^{10} + \frac{195143807603956877843}{1065829091546210290020} a^{9} - \frac{155583634969865226011}{1421105455394947053360} a^{8} - \frac{974665480399610019601}{4263316366184841160080} a^{7} - \frac{83449387969067031235}{852663273236968232016} a^{6} + \frac{285340891972678923347}{2131658183092420580040} a^{5} - \frac{601248629345364832277}{2131658183092420580040} a^{4} - \frac{455073679023498127651}{8526632732369682320160} a^{3} + \frac{5589547692042631033}{236850909232491175560} a^{2} + \frac{278995900800661681903}{710552727697473526680} a - \frac{52391936832302677889}{106582909154621029002}$

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:  $15$
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:  \( 7423969450.79 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T513:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 128 conjugacy class representatives for t20n513 are not computed
Character table for t20n513 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 5.5.2382032.1, 10.10.300726051798272.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$