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Properties

Label 20.0.21114116905...6201.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{10}\cdot 13^{10}$
Root discriminant $20.71$
Ramified primes $3, 11, 13$
Class number $4$
Class group $[4]$
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![352, 1056, 1952, 2012, 2680, 1161, 1188, -964, 2362, -1477, 1651, -233, 165, 16, 168, -54, 27, 3, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 + 3*x^17 + 27*x^16 - 54*x^15 + 168*x^14 + 16*x^13 + 165*x^12 - 233*x^11 + 1651*x^10 - 1477*x^9 + 2362*x^8 - 964*x^7 + 1188*x^6 + 1161*x^5 + 2680*x^4 + 2012*x^3 + 1952*x^2 + 1056*x + 352)
 
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 + 3*x^17 + 27*x^16 - 54*x^15 + 168*x^14 + 16*x^13 + 165*x^12 - 233*x^11 + 1651*x^10 - 1477*x^9 + 2362*x^8 - 964*x^7 + 1188*x^6 + 1161*x^5 + 2680*x^4 + 2012*x^3 + 1952*x^2 + 1056*x + 352, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 3 x^{18} + 3 x^{17} + 27 x^{16} - 54 x^{15} + 168 x^{14} + 16 x^{13} + 165 x^{12} - 233 x^{11} + 1651 x^{10} - 1477 x^{9} + 2362 x^{8} - 964 x^{7} + 1188 x^{6} + 1161 x^{5} + 2680 x^{4} + 2012 x^{3} + 1952 x^{2} + 1056 x + 352 \)

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:  \(211141169056178733575976201=3^{10}\cdot 11^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.71$
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:  $3, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{9} - \frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{1}{10} a^{5} + \frac{1}{10} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{10} a^{14} + \frac{1}{5} a^{11} + \frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{2}{5} a^{8} + \frac{1}{10} a^{7} - \frac{3}{10} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{10} a^{15} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{3}{10} a^{9} - \frac{1}{5} a^{8} - \frac{1}{2} a^{7} - \frac{3}{10} a^{6} + \frac{1}{10} a^{5} - \frac{3}{10} a^{4} - \frac{1}{10} a^{3} + \frac{3}{10} a^{2} + \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{110} a^{16} + \frac{2}{55} a^{15} - \frac{3}{110} a^{14} + \frac{1}{22} a^{13} + \frac{1}{55} a^{11} - \frac{12}{55} a^{10} + \frac{5}{11} a^{9} + \frac{23}{55} a^{8} - \frac{19}{55} a^{7} + \frac{9}{110} a^{6} - \frac{27}{55} a^{5} - \frac{49}{110} a^{4} - \frac{23}{55} a^{3} + \frac{2}{11} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{1100} a^{17} + \frac{47}{1100} a^{15} + \frac{39}{1100} a^{14} + \frac{13}{1100} a^{13} + \frac{1}{550} a^{12} - \frac{52}{275} a^{11} + \frac{117}{550} a^{10} - \frac{3}{20} a^{9} - \frac{13}{1100} a^{8} - \frac{367}{1100} a^{7} + \frac{427}{1100} a^{6} - \frac{219}{550} a^{5} + \frac{207}{550} a^{4} - \frac{151}{550} a^{3} - \frac{3}{1100} a^{2} + \frac{11}{25} a + \frac{1}{25}$, $\frac{1}{24200} a^{18} - \frac{1}{2420} a^{17} + \frac{67}{24200} a^{16} + \frac{1079}{24200} a^{15} + \frac{3}{24200} a^{14} + \frac{41}{12100} a^{13} + \frac{54}{3025} a^{12} + \frac{23}{275} a^{11} + \frac{833}{4840} a^{10} + \frac{11987}{24200} a^{9} + \frac{9483}{24200} a^{8} - \frac{513}{24200} a^{7} + \frac{5051}{12100} a^{6} + \frac{423}{3025} a^{5} - \frac{389}{3025} a^{4} - \frac{8243}{24200} a^{3} + \frac{2621}{6050} a^{2} - \frac{42}{275} a - \frac{6}{55}$, $\frac{1}{26926344164122444186970000} a^{19} + \frac{29436514128983695897}{2692634416412244418697000} a^{18} - \frac{7337671037470908272257}{26926344164122444186970000} a^{17} - \frac{641534414052540275153}{1583902597889555540410000} a^{16} + \frac{149184403702328874034631}{5385268832824488837394000} a^{15} + \frac{348706285844615033092553}{13463172082061222093485000} a^{14} - \frac{7378567818823437460623}{269263441641224441869700} a^{13} - \frac{52517702268905250731074}{1682896510257652761685625} a^{12} - \frac{6205243997202313963884483}{26926344164122444186970000} a^{11} + \frac{3444561391626657828230291}{26926344164122444186970000} a^{10} + \frac{7671163559375839335771803}{26926344164122444186970000} a^{9} + \frac{223296124525014621187667}{1583902597889555540410000} a^{8} - \frac{109796354728726674878603}{2692634416412244418697000} a^{7} - \frac{13755847912007360851774}{152990591841604796516875} a^{6} + \frac{250016922396338267217483}{1346317208206122209348500} a^{5} - \frac{1908197682363738690515319}{26926344164122444186970000} a^{4} - \frac{777526838129875895853097}{6731586041030611046742500} a^{3} + \frac{948711241843653037892347}{3365793020515305523371250} a^{2} + \frac{3134950597460688616923}{305981183683209593033750} a - \frac{6124234936978835889766}{152990591841604796516875}$

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

Class group and class number

$C_{4}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 35832.5653902 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-143}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{33}, \sqrt{-39})\), 5.1.20449.1 x5, 10.0.59797108943.2, 10.0.1320972497559.1 x5, 10.2.1117745959473.1 x5

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$