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Properties

Label 20.0.87887174756...7249.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 131^{10}$
Root discriminant $19.82$
Ramified primes $3, 131$
Class number $1$
Class group Trivial
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![16, -8, -160, -66, 1629, -58, -428, 434, -237, -176, 236, -126, -16, -38, -20, 50, -9, -8, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 12*x^18 - 8*x^17 - 9*x^16 + 50*x^15 - 20*x^14 - 38*x^13 - 16*x^12 - 126*x^11 + 236*x^10 - 176*x^9 - 237*x^8 + 434*x^7 - 428*x^6 - 58*x^5 + 1629*x^4 - 66*x^3 - 160*x^2 - 8*x + 16)
 
gp: K = bnfinit(x^20 - 4*x^19 + 12*x^18 - 8*x^17 - 9*x^16 + 50*x^15 - 20*x^14 - 38*x^13 - 16*x^12 - 126*x^11 + 236*x^10 - 176*x^9 - 237*x^8 + 434*x^7 - 428*x^6 - 58*x^5 + 1629*x^4 - 66*x^3 - 160*x^2 - 8*x + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 12 x^{18} - 8 x^{17} - 9 x^{16} + 50 x^{15} - 20 x^{14} - 38 x^{13} - 16 x^{12} - 126 x^{11} + 236 x^{10} - 176 x^{9} - 237 x^{8} + 434 x^{7} - 428 x^{6} - 58 x^{5} + 1629 x^{4} - 66 x^{3} - 160 x^{2} - 8 x + 16 \)

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:  \(87887174756230199195247249=3^{10}\cdot 131^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.82$
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:  $3, 131$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} + \frac{1}{16} a^{11} - \frac{3}{32} a^{10} - \frac{3}{32} a^{9} - \frac{1}{32} a^{7} - \frac{7}{32} a^{6} + \frac{3}{16} a^{5} - \frac{5}{32} a^{4} - \frac{13}{32} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{32} a^{14} + \frac{1}{32} a^{12} - \frac{1}{32} a^{11} + \frac{1}{16} a^{10} - \frac{3}{32} a^{9} - \frac{1}{32} a^{8} - \frac{1}{4} a^{7} - \frac{1}{32} a^{6} + \frac{1}{32} a^{5} + \frac{3}{16} a^{4} + \frac{11}{32} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{32} a^{15} + \frac{1}{16} a^{9} - \frac{1}{4} a^{6} + \frac{13}{32} a^{3} - \frac{1}{4}$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{128} a^{17} - \frac{1}{128} a^{16} - \frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{1}{64} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} + \frac{5}{64} a^{8} - \frac{11}{64} a^{7} + \frac{3}{16} a^{6} + \frac{15}{128} a^{5} + \frac{1}{128} a^{4} + \frac{17}{64} a^{3} - \frac{5}{16} a^{2} + \frac{3}{16} a - \frac{1}{8}$, $\frac{1}{13568} a^{18} - \frac{7}{3392} a^{17} + \frac{161}{13568} a^{16} + \frac{43}{3392} a^{15} - \frac{37}{3392} a^{14} + \frac{71}{6784} a^{13} - \frac{7}{424} a^{12} + \frac{23}{1696} a^{11} - \frac{1}{212} a^{10} + \frac{713}{6784} a^{9} - \frac{23}{3392} a^{8} - \frac{1307}{6784} a^{7} - \frac{1745}{13568} a^{6} + \frac{121}{3392} a^{5} - \frac{833}{13568} a^{4} + \frac{1697}{6784} a^{3} - \frac{259}{848} a^{2} + \frac{421}{1696} a + \frac{227}{848}$, $\frac{1}{98849109315819008} a^{19} + \frac{1230971153341}{98849109315819008} a^{18} - \frac{189956035007707}{98849109315819008} a^{17} + \frac{145595616380285}{98849109315819008} a^{16} - \frac{81553407643539}{6178069332238688} a^{15} - \frac{451435361999}{49424554657909504} a^{14} + \frac{646819133640151}{49424554657909504} a^{13} - \frac{317478621549869}{6178069332238688} a^{12} + \frac{578572674030945}{6178069332238688} a^{11} - \frac{4624848680600543}{49424554657909504} a^{10} - \frac{4417952683044745}{49424554657909504} a^{9} + \frac{3791331962889955}{49424554657909504} a^{8} - \frac{13209597532818631}{98849109315819008} a^{7} - \frac{6844708089386333}{98849109315819008} a^{6} + \frac{295817258967561}{1190953124286976} a^{5} + \frac{20145247231287457}{98849109315819008} a^{4} - \frac{3103363368093071}{49424554657909504} a^{3} + \frac{210108207292979}{426073747050944} a^{2} - \frac{2195414261346229}{12356138664477376} a - \frac{2310129109011685}{6178069332238688}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{6102865552907}{1865077534260736} a^{19} + \frac{12872332362887}{1865077534260736} a^{18} - \frac{27876815375491}{1865077534260736} a^{17} - \frac{86140723279301}{1865077534260736} a^{16} + \frac{8455634514415}{116567345891296} a^{15} - \frac{92819113258147}{932538767130368} a^{14} - \frac{233920514898783}{932538767130368} a^{13} + \frac{27702733797081}{116567345891296} a^{12} + \frac{8829491819405}{29141836472824} a^{11} + \frac{462335467653045}{932538767130368} a^{10} + \frac{55941330005705}{932538767130368} a^{9} - \frac{799376282355673}{932538767130368} a^{8} + \frac{3448832028149953}{1865077534260736} a^{7} + \frac{325651953256921}{1865077534260736} a^{6} - \frac{31280466536143}{22470813665792} a^{5} + \frac{5485935105792763}{1865077534260736} a^{4} - \frac{4530499135485425}{932538767130368} a^{3} - \frac{82350658983441}{8039127302848} a^{2} + \frac{104896237584745}{233134691782592} a + \frac{122705779918309}{116567345891296} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1203172.53469 \)
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

\(\Q(\sqrt{-131}) \), \(\Q(\sqrt{393}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-131})\), 5.1.17161.1 x5, 10.0.38579489651.1, 10.2.9374815985193.1 x5, 10.0.71563480803.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
$131$131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$