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Properties

Label 20.0.14469785316...5744.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 53^{14}$
Root discriminant $32.21$
Ramified primes $2, 53$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![153, 624, 542, -1902, -1014, 2560, 1168, -4158, 2754, 1380, -3735, 2794, -612, -774, 992, -686, 346, -128, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 36*x^18 - 128*x^17 + 346*x^16 - 686*x^15 + 992*x^14 - 774*x^13 - 612*x^12 + 2794*x^11 - 3735*x^10 + 1380*x^9 + 2754*x^8 - 4158*x^7 + 1168*x^6 + 2560*x^5 - 1014*x^4 - 1902*x^3 + 542*x^2 + 624*x + 153)
 
gp: K = bnfinit(x^20 - 8*x^19 + 36*x^18 - 128*x^17 + 346*x^16 - 686*x^15 + 992*x^14 - 774*x^13 - 612*x^12 + 2794*x^11 - 3735*x^10 + 1380*x^9 + 2754*x^8 - 4158*x^7 + 1168*x^6 + 2560*x^5 - 1014*x^4 - 1902*x^3 + 542*x^2 + 624*x + 153, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 36 x^{18} - 128 x^{17} + 346 x^{16} - 686 x^{15} + 992 x^{14} - 774 x^{13} - 612 x^{12} + 2794 x^{11} - 3735 x^{10} + 1380 x^{9} + 2754 x^{8} - 4158 x^{7} + 1168 x^{6} + 2560 x^{5} - 1014 x^{4} - 1902 x^{3} + 542 x^{2} + 624 x + 153 \)

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:  \(1446978531682831584079714975744=2^{20}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.21$
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:  $2, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{123} a^{17} - \frac{8}{123} a^{16} + \frac{11}{123} a^{15} - \frac{5}{41} a^{14} + \frac{29}{123} a^{13} + \frac{1}{41} a^{12} + \frac{14}{123} a^{11} + \frac{31}{123} a^{10} + \frac{7}{123} a^{9} + \frac{22}{123} a^{8} + \frac{35}{123} a^{7} + \frac{19}{41} a^{6} + \frac{2}{41} a^{5} + \frac{23}{123} a^{4} + \frac{12}{41} a^{3} - \frac{18}{41} a^{2} + \frac{7}{41} a + \frac{2}{41}$, $\frac{1}{6519} a^{18} + \frac{7}{2173} a^{17} - \frac{877}{6519} a^{16} + \frac{1042}{6519} a^{15} - \frac{611}{6519} a^{14} - \frac{607}{2173} a^{13} - \frac{841}{2173} a^{12} + \frac{9}{2173} a^{11} + \frac{2341}{6519} a^{10} - \frac{2317}{6519} a^{9} + \frac{853}{2173} a^{8} + \frac{662}{6519} a^{7} + \frac{1454}{6519} a^{6} + \frac{238}{6519} a^{5} - \frac{1064}{2173} a^{4} + \frac{1277}{6519} a^{3} + \frac{382}{6519} a^{2} + \frac{55}{159} a + \frac{304}{2173}$, $\frac{1}{673330795240187622020152347} a^{19} + \frac{22181988927499705225357}{673330795240187622020152347} a^{18} + \frac{759734518528976295195820}{224443598413395874006717449} a^{17} + \frac{105966724233498252025245736}{673330795240187622020152347} a^{16} - \frac{90665588427095733034888943}{673330795240187622020152347} a^{15} - \frac{62260735792738612014648638}{673330795240187622020152347} a^{14} - \frac{181279352064192353006631736}{673330795240187622020152347} a^{13} + \frac{45416381745361901195715109}{224443598413395874006717449} a^{12} - \frac{32883003780847416900990835}{224443598413395874006717449} a^{11} - \frac{19368298586861418949176302}{673330795240187622020152347} a^{10} - \frac{70226806551685464653400007}{224443598413395874006717449} a^{9} + \frac{2017297686200703316107634}{4234784875724450452956933} a^{8} - \frac{32531258358263396254138983}{74814532804465291335572483} a^{7} + \frac{11820994012590724131617873}{224443598413395874006717449} a^{6} + \frac{56673863428622202471803662}{673330795240187622020152347} a^{5} - \frac{193945197502253241137138612}{673330795240187622020152347} a^{4} + \frac{20060150634795355071597745}{74814532804465291335572483} a^{3} + \frac{4436422334884483512545061}{74814532804465291335572483} a^{2} + \frac{95437711480947434166240371}{673330795240187622020152347} a + \frac{76436319729821383129842365}{224443598413395874006717449}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{9771247905794102}{4496688501934984371} a^{19} - \frac{85981938580996600}{4496688501934984371} a^{18} + \frac{137296221640318966}{1498896167311661457} a^{17} - \frac{1515555019964441038}{4496688501934984371} a^{16} + \frac{4315341215380311617}{4496688501934984371} a^{15} - \frac{9187895945902766416}{4496688501934984371} a^{14} + \frac{14528212489567384348}{4496688501934984371} a^{13} - \frac{4818753443075223727}{1498896167311661457} a^{12} - \frac{290520465355069993}{1498896167311661457} a^{11} + \frac{32091374658823631702}{4496688501934984371} a^{10} - \frac{6231291731679425208}{499632055770553819} a^{9} + \frac{4340524368285863664}{499632055770553819} a^{8} + \frac{5704370490953934985}{1498896167311661457} a^{7} - \frac{18787684468924276559}{1498896167311661457} a^{6} + \frac{35216842558375670618}{4496688501934984371} a^{5} + \frac{18439340900833984214}{4496688501934984371} a^{4} - \frac{2523089485617295884}{499632055770553819} a^{3} - \frac{2310414523356107875}{499632055770553819} a^{2} + \frac{16701053283457885345}{4496688501934984371} a + \frac{1934483219623708516}{1498896167311661457} \) (order $4$)
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

$C_2\times F_5$ (as 20T13):

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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-53}) \), \(\Q(\sqrt{53}) \), \(\Q(i, \sqrt{53})\), 5.5.2382032.1, 10.0.22696305796096.1, 10.0.1202904207193088.1, 10.10.300726051798272.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ 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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$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}$
53Data not computed