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Properties

Label 20.0.13682086378...1936.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{2}\cdot 41^{2}\cdot 53^{14}$
Root discriminant $45.37$
Ramified primes $2, 3, 41, 53$
Class number $120$ (GRH)
Class group $[120]$ (GRH)
Galois group 20T140

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -2048, 192, 6848, 6580, -5772, 14483, -3541, -4748, 5343, -1837, -1698, 2557, -817, 43, -202, 57, 3, 16, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 16*x^18 + 3*x^17 + 57*x^16 - 202*x^15 + 43*x^14 - 817*x^13 + 2557*x^12 - 1698*x^11 - 1837*x^10 + 5343*x^9 - 4748*x^8 - 3541*x^7 + 14483*x^6 - 5772*x^5 + 6580*x^4 + 6848*x^3 + 192*x^2 - 2048*x + 1024)
 
gp: K = bnfinit(x^20 - 5*x^19 + 16*x^18 + 3*x^17 + 57*x^16 - 202*x^15 + 43*x^14 - 817*x^13 + 2557*x^12 - 1698*x^11 - 1837*x^10 + 5343*x^9 - 4748*x^8 - 3541*x^7 + 14483*x^6 - 5772*x^5 + 6580*x^4 + 6848*x^3 + 192*x^2 - 2048*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 16 x^{18} + 3 x^{17} + 57 x^{16} - 202 x^{15} + 43 x^{14} - 817 x^{13} + 2557 x^{12} - 1698 x^{11} - 1837 x^{10} + 5343 x^{9} - 4748 x^{8} - 3541 x^{7} + 14483 x^{6} - 5772 x^{5} + 6580 x^{4} + 6848 x^{3} + 192 x^{2} - 2048 x + 1024 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1368208637864347439721375491751936=2^{16}\cdot 3^{2}\cdot 41^{2}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.37$
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:  $2, 3, 41, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{4} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{7} - \frac{1}{4} a^{5} + \frac{5}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{10} - \frac{1}{32} a^{8} - \frac{3}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} + \frac{9}{32} a^{4} - \frac{7}{16} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{5}{32} a^{5} + \frac{1}{16} a^{4} + \frac{5}{16} a^{3} + \frac{3}{8} a^{2}$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{13} - \frac{1}{64} a^{12} + \frac{1}{64} a^{10} - \frac{1}{64} a^{9} + \frac{3}{64} a^{8} + \frac{3}{16} a^{7} + \frac{9}{64} a^{6} + \frac{11}{64} a^{5} + \frac{15}{64} a^{4} + \frac{5}{16} a^{3} - \frac{5}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{128} a^{15} - \frac{1}{128} a^{14} + \frac{1}{128} a^{13} - \frac{1}{64} a^{12} + \frac{1}{128} a^{11} + \frac{3}{128} a^{10} + \frac{1}{128} a^{9} + \frac{1}{64} a^{8} + \frac{21}{128} a^{7} - \frac{21}{128} a^{6} + \frac{21}{128} a^{5} - \frac{29}{64} a^{4} - \frac{15}{32} a^{3} - \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{20352} a^{16} + \frac{11}{5088} a^{15} - \frac{41}{10176} a^{14} - \frac{69}{6784} a^{13} + \frac{77}{20352} a^{12} + \frac{1}{848} a^{11} - \frac{223}{10176} a^{10} - \frac{137}{6784} a^{9} + \frac{845}{20352} a^{8} - \frac{1033}{5088} a^{7} + \frac{581}{3392} a^{6} - \frac{1163}{20352} a^{5} + \frac{125}{2544} a^{4} + \frac{2419}{5088} a^{3} + \frac{257}{1272} a^{2} + \frac{25}{636} a - \frac{50}{159}$, $\frac{1}{40704} a^{17} + \frac{49}{40704} a^{15} - \frac{1}{159} a^{14} + \frac{61}{20352} a^{13} - \frac{23}{5088} a^{12} + \frac{565}{40704} a^{11} + \frac{275}{5088} a^{10} + \frac{1117}{20352} a^{9} - \frac{235}{5088} a^{8} - \frac{5347}{40704} a^{7} - \frac{467}{2544} a^{6} - \frac{4591}{40704} a^{5} - \frac{109}{636} a^{4} + \frac{1169}{3392} a^{3} + \frac{143}{318} a^{2} + \frac{145}{636} a - \frac{13}{159}$, $\frac{1}{162816} a^{18} - \frac{1}{54272} a^{16} - \frac{1}{256} a^{15} + \frac{201}{27136} a^{14} + \frac{289}{20352} a^{13} - \frac{895}{162816} a^{12} - \frac{1193}{40704} a^{11} - \frac{2869}{81408} a^{10} - \frac{823}{20352} a^{9} + \frac{3075}{54272} a^{8} + \frac{9077}{40704} a^{7} - \frac{16051}{162816} a^{6} - \frac{287}{2544} a^{5} - \frac{13309}{40704} a^{4} + \frac{175}{424} a^{3} - \frac{163}{636} a^{2} - \frac{8}{53} a + \frac{7}{159}$, $\frac{1}{13599163523040406318080} a^{19} - \frac{30709542833611837}{13599163523040406318080} a^{18} - \frac{5205216728379761}{906610901536027087872} a^{17} + \frac{22276379135163627}{1511018169226711813120} a^{16} - \frac{2641898719922964139}{2266527253840067719680} a^{15} + \frac{49659520839293363233}{6799581761520203159040} a^{14} + \frac{6963526720976620249}{1511018169226711813120} a^{13} + \frac{9499523371623355031}{13599163523040406318080} a^{12} + \frac{2249640483904816791}{151101816922671181312} a^{11} - \frac{48957557112606531413}{2266527253840067719680} a^{10} - \frac{235880746706307923191}{13599163523040406318080} a^{9} - \frac{145898366004094176869}{2719832704608081263616} a^{8} + \frac{325443424765584342893}{1511018169226711813120} a^{7} - \frac{194818461447396498413}{2719832704608081263616} a^{6} + \frac{41856865343549611693}{849947720190025394880} a^{5} - \frac{176569448948455926107}{3399790880760101579520} a^{4} + \frac{341623745437389689891}{849947720190025394880} a^{3} + \frac{89388975299924989969}{212486930047506348720} a^{2} - \frac{2090514956019263081}{5312173251187658718} a + \frac{6414890661088758083}{13280433127969146795}$

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

Class group and class number

$C_{120}$, which has order $120$ (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:  $9$
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:  \( 7826958.70528 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T140:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 22 conjugacy class representatives for t20n140
Character table for t20n140 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 5.5.2382032.1, 10.0.36989304371187456.1, 10.0.697911403229952.1, 10.10.300726051798272.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ 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}$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
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}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
53Data not computed