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Properties

Label 20.4.12103554505...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{18}\cdot 5^{14}\cdot 13^{15}$
Root discriminant $56.77$
Ramified primes $3, 5, 13$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6379, 5873, 56785, 41172, -10179, 132846, -67458, -285510, -30831, 53630, -638, -10390, 4821, -855, -723, 582, -111, -24, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 24*x^17 - 111*x^16 + 582*x^15 - 723*x^14 - 855*x^13 + 4821*x^12 - 10390*x^11 - 638*x^10 + 53630*x^9 - 30831*x^8 - 285510*x^7 - 67458*x^6 + 132846*x^5 - 10179*x^4 + 41172*x^3 + 56785*x^2 + 5873*x + 6379)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 - 24*x^17 - 111*x^16 + 582*x^15 - 723*x^14 - 855*x^13 + 4821*x^12 - 10390*x^11 - 638*x^10 + 53630*x^9 - 30831*x^8 - 285510*x^7 - 67458*x^6 + 132846*x^5 - 10179*x^4 + 41172*x^3 + 56785*x^2 + 5873*x + 6379, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} - 24 x^{17} - 111 x^{16} + 582 x^{15} - 723 x^{14} - 855 x^{13} + 4821 x^{12} - 10390 x^{11} - 638 x^{10} + 53630 x^{9} - 30831 x^{8} - 285510 x^{7} - 67458 x^{6} + 132846 x^{5} - 10179 x^{4} + 41172 x^{3} + 56785 x^{2} + 5873 x + 6379 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(121035545052616729536866290283203125=3^{18}\cdot 5^{14}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.77$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{11} + \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{6} a^{9} + \frac{1}{4} a^{6} + \frac{1}{3} a^{4} - \frac{1}{12} a^{3} - \frac{1}{12} a^{2} + \frac{1}{6} a - \frac{5}{12}$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{13} + \frac{1}{36} a^{12} + \frac{1}{36} a^{11} + \frac{1}{18} a^{10} - \frac{1}{18} a^{9} + \frac{1}{3} a^{8} - \frac{5}{12} a^{7} + \frac{1}{6} a^{6} - \frac{1}{18} a^{5} - \frac{7}{36} a^{4} - \frac{17}{36} a^{3} - \frac{7}{18} a^{2} + \frac{11}{36} a + \frac{5}{18}$, $\frac{1}{36} a^{15} + \frac{1}{18} a^{12} - \frac{1}{12} a^{11} - \frac{1}{18} a^{9} + \frac{5}{12} a^{8} + \frac{1}{4} a^{7} + \frac{1}{9} a^{6} + \frac{1}{4} a^{5} + \frac{1}{3} a^{4} - \frac{13}{36} a^{3} - \frac{1}{4} a^{2} + \frac{1}{12} a + \frac{4}{9}$, $\frac{1}{72} a^{16} - \frac{1}{72} a^{15} - \frac{1}{72} a^{14} - \frac{1}{24} a^{12} - \frac{1}{72} a^{11} - \frac{1}{72} a^{10} + \frac{13}{72} a^{9} - \frac{1}{2} a^{8} - \frac{1}{9} a^{7} - \frac{5}{36} a^{6} + \frac{23}{72} a^{5} + \frac{1}{12} a^{4} + \frac{1}{12} a^{3} - \frac{13}{72} a^{2} + \frac{7}{36} a + \frac{31}{72}$, $\frac{1}{216} a^{17} - \frac{1}{216} a^{16} + \frac{1}{216} a^{15} + \frac{1}{108} a^{14} + \frac{7}{216} a^{13} + \frac{5}{216} a^{12} - \frac{5}{216} a^{11} + \frac{29}{216} a^{10} - \frac{10}{27} a^{9} - \frac{13}{108} a^{8} + \frac{7}{108} a^{7} + \frac{43}{216} a^{6} - \frac{2}{27} a^{5} + \frac{7}{54} a^{4} + \frac{107}{216} a^{3} + \frac{1}{54} a^{2} + \frac{35}{216} a - \frac{5}{54}$, $\frac{1}{432} a^{18} - \frac{1}{144} a^{16} - \frac{1}{72} a^{14} + \frac{1}{36} a^{13} + \frac{11}{144} a^{12} - \frac{1}{16} a^{11} + \frac{1}{72} a^{10} - \frac{97}{432} a^{9} - \frac{17}{72} a^{8} + \frac{3}{16} a^{7} + \frac{65}{144} a^{6} - \frac{25}{144} a^{5} - \frac{1}{48} a^{4} - \frac{13}{48} a^{3} + \frac{1}{72} a^{2} - \frac{11}{48} a - \frac{11}{432}$, $\frac{1}{57745863525119499248129374700926615165717824} a^{19} - \frac{4184160234828443741662193269295699935097}{6416207058346611027569930522325179462857536} a^{18} - \frac{18911498056020325274377839813225936574531}{19248621175039833082709791566975538388572608} a^{17} - \frac{31128192696328364039682217504121079177865}{19248621175039833082709791566975538388572608} a^{16} - \frac{4944211580824632002308597773060175123069}{401012941146663189223120657645323716428596} a^{15} + \frac{24289585163553216075191326037460781660057}{9624310587519916541354895783487769194286304} a^{14} - \frac{721003477163446763198669964854880850374595}{19248621175039833082709791566975538388572608} a^{13} - \frac{11511065546631735520563001025207361996335}{9624310587519916541354895783487769194286304} a^{12} + \frac{129896571259499529604804355778001138475231}{6416207058346611027569930522325179462857536} a^{11} + \frac{1976414894941338820096212026326316553478463}{57745863525119499248129374700926615165717824} a^{10} + \frac{7851680419853629093264939269961366327017269}{19248621175039833082709791566975538388572608} a^{9} + \frac{440399448671292893319024268418134981936089}{2138735686115537009189976840775059820952512} a^{8} - \frac{950653474958569861064899133594794473122089}{9624310587519916541354895783487769194286304} a^{7} + \frac{971566602543352902044927021943017984364635}{2406077646879979135338723945871942298571576} a^{6} - \frac{2505046313272951364078775913886227067279563}{9624310587519916541354895783487769194286304} a^{5} + \frac{10376345307541947019950894884602299077098}{33417745095555265768593388137110309702383} a^{4} + \frac{7571183121579640390416386081623207929952591}{19248621175039833082709791566975538388572608} a^{3} - \frac{6174560629149942780843954767233210146667055}{19248621175039833082709791566975538388572608} a^{2} + \frac{1028625492116448171164944135565218745731337}{28872931762559749624064687350463307582858912} a - \frac{115223096105994278776441959329575357608371}{2138735686115537009189976840775059820952512}$

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:  $11$
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:  \( 2292437913.7392983 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

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 $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.494325.1, 5.1.1711125.1, 10.2.38063333953125.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ $20$

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$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
5Data not computed
13Data not computed