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Properties

Label 20.0.14658323082...2757.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 13^{5}\cdot 401^{8}$
Root discriminant $36.17$
Ramified primes $3, 13, 401$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $D_4\times D_5$ (as 20T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -21, -56, 310, 121, -1201, -246, 2615, 827, -2628, -814, 1656, 388, -609, -106, 140, 15, -16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 16*x^18 + 15*x^17 + 140*x^16 - 106*x^15 - 609*x^14 + 388*x^13 + 1656*x^12 - 814*x^11 - 2628*x^10 + 827*x^9 + 2615*x^8 - 246*x^7 - 1201*x^6 + 121*x^5 + 310*x^4 - 56*x^3 - 21*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 16*x^18 + 15*x^17 + 140*x^16 - 106*x^15 - 609*x^14 + 388*x^13 + 1656*x^12 - 814*x^11 - 2628*x^10 + 827*x^9 + 2615*x^8 - 246*x^7 - 1201*x^6 + 121*x^5 + 310*x^4 - 56*x^3 - 21*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 16 x^{18} + 15 x^{17} + 140 x^{16} - 106 x^{15} - 609 x^{14} + 388 x^{13} + 1656 x^{12} - 814 x^{11} - 2628 x^{10} + 827 x^{9} + 2615 x^{8} - 246 x^{7} - 1201 x^{6} + 121 x^{5} + 310 x^{4} - 56 x^{3} - 21 x^{2} + 3 x + 1 \)

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:  \(14658323082243334363275748982757=3^{10}\cdot 13^{5}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.17$
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:  $3, 13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{2}{9} a^{12} + \frac{1}{3} a^{11} + \frac{2}{9} a^{10} - \frac{2}{9} a^{9} + \frac{4}{9} a^{8} + \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{15} - \frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{4}{9} a^{12} - \frac{1}{9} a^{11} - \frac{4}{9} a^{10} - \frac{1}{3} a^{9} - \frac{4}{9} a^{8} + \frac{1}{9} a^{7} + \frac{4}{9} a^{6} - \frac{2}{9} a^{5} + \frac{4}{9} a^{4} + \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{272367} a^{18} - \frac{2300}{272367} a^{17} + \frac{9251}{272367} a^{16} + \frac{6886}{272367} a^{15} + \frac{16216}{272367} a^{14} - \frac{122003}{272367} a^{13} - \frac{101414}{272367} a^{12} + \frac{131521}{272367} a^{11} - \frac{108991}{272367} a^{10} + \frac{72674}{272367} a^{9} - \frac{539}{1713} a^{8} + \frac{96395}{272367} a^{7} + \frac{117653}{272367} a^{6} + \frac{60362}{272367} a^{5} - \frac{73630}{272367} a^{4} - \frac{116623}{272367} a^{3} - \frac{38893}{90789} a^{2} + \frac{11392}{30263} a + \frac{68410}{272367}$, $\frac{1}{45694975365189} a^{19} + \frac{32020141}{45694975365189} a^{18} - \frac{1306601693285}{45694975365189} a^{17} + \frac{1206487187066}{45694975365189} a^{16} - \frac{113787424594}{801666234477} a^{15} - \frac{6466126871662}{45694975365189} a^{14} + \frac{303764927923}{2404998703431} a^{13} + \frac{1572301346977}{15231658455063} a^{12} + \frac{4956975949204}{15231658455063} a^{11} + \frac{7377557600276}{45694975365189} a^{10} + \frac{1997982611}{862169346513} a^{9} - \frac{9151573685342}{45694975365189} a^{8} - \frac{471469228997}{45694975365189} a^{7} + \frac{8278989162671}{45694975365189} a^{6} - \frac{766121046531}{1692406495007} a^{5} - \frac{15387787960670}{45694975365189} a^{4} + \frac{5304847046957}{15231658455063} a^{3} + \frac{13817367459544}{45694975365189} a^{2} - \frac{13220985449410}{45694975365189} a + \frac{266537284586}{862169346513}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{118903}{272367} a^{19} - \frac{94222}{272367} a^{18} - \frac{1932344}{272367} a^{17} + \frac{155988}{30263} a^{16} + \frac{17080954}{272367} a^{15} - \frac{9375335}{272367} a^{14} - \frac{75466706}{272367} a^{13} + \frac{32878618}{272367} a^{12} + \frac{207394469}{272367} a^{11} - \frac{20890214}{90789} a^{10} - \frac{332398558}{272367} a^{9} + \frac{48841120}{272367} a^{8} + \frac{6124150}{5139} a^{7} + \frac{17382971}{272367} a^{6} - \frac{137442280}{272367} a^{5} - \frac{3666115}{272367} a^{4} + \frac{10181270}{90789} a^{3} - \frac{3063896}{272367} a^{2} - \frac{322271}{90789} a + \frac{136604}{272367} \) (order $6$)
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:  \( 19881261.2295 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times D_5$ (as 20T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $D_4\times D_5$
Character table for $D_4\times D_5$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.117.1, 5.5.160801.1, 10.0.6283241669043.1

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 $20$ R $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ $20$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_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}$
401Data not computed