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Properties

Label 20.0.23376400555...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{15}\cdot 43^{8}$
Root discriminant $26.21$
Ramified primes $2, 5, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 50, 50, -125, 565, 110, 3820, -870, 4296, -854, 2316, -786, 897, -476, 416, -233, 134, -48, 18, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 18*x^18 - 48*x^17 + 134*x^16 - 233*x^15 + 416*x^14 - 476*x^13 + 897*x^12 - 786*x^11 + 2316*x^10 - 854*x^9 + 4296*x^8 - 870*x^7 + 3820*x^6 + 110*x^5 + 565*x^4 - 125*x^3 + 50*x^2 + 50*x + 25)
 
gp: K = bnfinit(x^20 - 5*x^19 + 18*x^18 - 48*x^17 + 134*x^16 - 233*x^15 + 416*x^14 - 476*x^13 + 897*x^12 - 786*x^11 + 2316*x^10 - 854*x^9 + 4296*x^8 - 870*x^7 + 3820*x^6 + 110*x^5 + 565*x^4 - 125*x^3 + 50*x^2 + 50*x + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 18 x^{18} - 48 x^{17} + 134 x^{16} - 233 x^{15} + 416 x^{14} - 476 x^{13} + 897 x^{12} - 786 x^{11} + 2316 x^{10} - 854 x^{9} + 4296 x^{8} - 870 x^{7} + 3820 x^{6} + 110 x^{5} + 565 x^{4} - 125 x^{3} + 50 x^{2} + 50 x + 25 \)

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:  \(23376400555202000000000000000=2^{16}\cdot 5^{15}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.21$
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:  $2, 5, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{9} - \frac{2}{5} a^{4}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{10} - \frac{2}{5} a^{5}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{11} - \frac{2}{5} a^{6}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{315} a^{18} - \frac{1}{63} a^{17} + \frac{23}{315} a^{16} - \frac{2}{63} a^{15} - \frac{1}{105} a^{14} - \frac{31}{315} a^{13} + \frac{23}{315} a^{12} + \frac{25}{63} a^{11} + \frac{4}{315} a^{10} + \frac{4}{45} a^{9} + \frac{131}{315} a^{8} + \frac{1}{3} a^{7} + \frac{20}{63} a^{6} + \frac{32}{105} a^{5} + \frac{11}{35} a^{4} - \frac{8}{63} a^{3} + \frac{23}{63} a^{2} - \frac{2}{63} a - \frac{1}{63}$, $\frac{1}{474575388958690193757791928825} a^{19} - \frac{651323061868921485852648}{1506588536376794265897752155} a^{18} - \frac{41453471248437904441700721662}{474575388958690193757791928825} a^{17} - \frac{1191088287159553944455218873}{22598828045651913988466282325} a^{16} + \frac{25827836699437613125250153464}{474575388958690193757791928825} a^{15} + \frac{42296830916473891779862727267}{474575388958690193757791928825} a^{14} + \frac{14618983156643726469245976127}{158191796319563397919263976275} a^{13} - \frac{7591138102740489774824388892}{158191796319563397919263976275} a^{12} - \frac{179642198155825969215657055093}{474575388958690193757791928825} a^{11} - \frac{31114725730135031796709219292}{158191796319563397919263976275} a^{10} + \frac{87286180878190522252742528431}{474575388958690193757791928825} a^{9} + \frac{75905093874292182074092214596}{474575388958690193757791928825} a^{8} - \frac{149072430364779789043135364039}{474575388958690193757791928825} a^{7} - \frac{35188159299036287853709108202}{94915077791738038751558385765} a^{6} + \frac{15605098747161455144945872454}{31638359263912679583852795255} a^{5} - \frac{5633135822961195015278096951}{13559296827391148393079769395} a^{4} + \frac{17499752299305778712524946839}{94915077791738038751558385765} a^{3} - \frac{6784126526905219690375339313}{18983015558347607750311677153} a^{2} + \frac{2936100224852116816353769184}{18983015558347607750311677153} a + \frac{2865393160510437772608878036}{18983015558347607750311677153}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{230144309751139391357564854}{18983015558347607750311677153} a^{19} + \frac{6334113817724518113515052439}{94915077791738038751558385765} a^{18} - \frac{23497341087047573885839601929}{94915077791738038751558385765} a^{17} + \frac{65085573403083792282240490739}{94915077791738038751558385765} a^{16} - \frac{59975356407172980584744764352}{31638359263912679583852795255} a^{15} + \frac{340219141716575494287320320637}{94915077791738038751558385765} a^{14} - \frac{85447490479809924291918293743}{13559296827391148393079769395} a^{13} + \frac{152304407402958375572598226063}{18983015558347607750311677153} a^{12} - \frac{1259934116340599451448043055302}{94915077791738038751558385765} a^{11} + \frac{273687676056622236270293101370}{18983015558347607750311677153} a^{10} - \frac{3016057555598533981011312117424}{94915077791738038751558385765} a^{9} + \frac{148612486071999827936517477362}{6327671852782535916770559051} a^{8} - \frac{5172344919801252215170180482476}{94915077791738038751558385765} a^{7} + \frac{375731561445800917815301256296}{10546119754637559861284265085} a^{6} - \frac{492520190262245629374373722896}{10546119754637559861284265085} a^{5} + \frac{1934725066289119780355406284549}{94915077791738038751558385765} a^{4} - \frac{42809204352191124442586037964}{18983015558347607750311677153} a^{3} + \frac{85623782500824690792555259273}{18983015558347607750311677153} a^{2} - \frac{3268351038328220634073451449}{2711859365478229678615953879} a - \frac{1620636797498167006478469004}{6327671852782535916770559051} \) (order $10$)
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:  \( 3323311.71225 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.3698000.1, 10.10.68376020000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$43$43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$