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Properties

Label 20.0.12902360962...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{34}\cdot 17^{10}$
Root discriminant $254.41$
Ramified primes $2, 5, 17$
Class number $245209008$ (GRH)
Class group $[2, 122604504]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49402019223649, 0, 14887410527530, 0, 2056587912445, 0, 171855099720, 0, 9761928930, 0, 404258124, 0, 12753090, 0, 310760, 0, 5885, 0, 90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 90*x^18 + 5885*x^16 + 310760*x^14 + 12753090*x^12 + 404258124*x^10 + 9761928930*x^8 + 171855099720*x^6 + 2056587912445*x^4 + 14887410527530*x^2 + 49402019223649)
 
gp: K = bnfinit(x^20 + 90*x^18 + 5885*x^16 + 310760*x^14 + 12753090*x^12 + 404258124*x^10 + 9761928930*x^8 + 171855099720*x^6 + 2056587912445*x^4 + 14887410527530*x^2 + 49402019223649, 1)
 

Normalized defining polynomial

\( x^{20} + 90 x^{18} + 5885 x^{16} + 310760 x^{14} + 12753090 x^{12} + 404258124 x^{10} + 9761928930 x^{8} + 171855099720 x^{6} + 2056587912445 x^{4} + 14887410527530 x^{2} + 49402019223649 \)

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:  \(1290236096287360000000000000000000000000000000000=2^{40}\cdot 5^{34}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $254.41$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3400=2^{3}\cdot 5^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3400}(1,·)$, $\chi_{3400}(69,·)$, $\chi_{3400}(2311,·)$, $\chi_{3400}(2379,·)$, $\chi_{3400}(271,·)$, $\chi_{3400}(1361,·)$, $\chi_{3400}(339,·)$, $\chi_{3400}(1429,·)$, $\chi_{3400}(1631,·)$, $\chi_{3400}(2721,·)$, $\chi_{3400}(1699,·)$, $\chi_{3400}(2789,·)$, $\chi_{3400}(681,·)$, $\chi_{3400}(749,·)$, $\chi_{3400}(2991,·)$, $\chi_{3400}(3059,·)$, $\chi_{3400}(951,·)$, $\chi_{3400}(2041,·)$, $\chi_{3400}(1019,·)$, $\chi_{3400}(2109,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} + \frac{1}{16} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} + \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{1}{64} a^{9} - \frac{1}{64} a^{8} + \frac{1}{32} a^{7} - \frac{1}{32} a^{6} + \frac{1}{32} a^{5} - \frac{1}{32} a^{4} - \frac{3}{64} a^{3} + \frac{3}{64} a^{2} + \frac{29}{64} a - \frac{29}{64}$, $\frac{1}{448} a^{12} + \frac{1}{112} a^{10} + \frac{9}{448} a^{8} + \frac{3}{56} a^{6} - \frac{37}{448} a^{4} - \frac{23}{112} a^{2} - \frac{101}{448}$, $\frac{1}{896} a^{13} - \frac{1}{896} a^{12} + \frac{1}{224} a^{11} - \frac{1}{224} a^{10} - \frac{19}{896} a^{9} + \frac{19}{896} a^{8} + \frac{3}{112} a^{7} - \frac{3}{112} a^{6} - \frac{93}{896} a^{5} + \frac{93}{896} a^{4} + \frac{33}{224} a^{3} - \frac{33}{224} a^{2} + \frac{207}{896} a - \frac{207}{896}$, $\frac{1}{896} a^{14} - \frac{1}{896} a^{12} - \frac{11}{896} a^{10} - \frac{3}{128} a^{8} - \frac{45}{896} a^{6} - \frac{19}{896} a^{4} - \frac{201}{896} a^{2} + \frac{169}{896}$, $\frac{1}{1792} a^{15} - \frac{1}{1792} a^{14} - \frac{1}{1792} a^{13} + \frac{1}{1792} a^{12} - \frac{11}{1792} a^{11} + \frac{11}{1792} a^{10} + \frac{5}{256} a^{9} - \frac{5}{256} a^{8} + \frac{67}{1792} a^{7} - \frac{67}{1792} a^{6} + \frac{205}{1792} a^{5} - \frac{205}{1792} a^{4} + \frac{135}{1792} a^{3} - \frac{135}{1792} a^{2} - \frac{559}{1792} a + \frac{559}{1792}$, $\frac{1}{12544} a^{16} - \frac{1}{6272} a^{14} + \frac{5}{6272} a^{12} + \frac{9}{896} a^{10} + \frac{25}{3136} a^{8} - \frac{251}{6272} a^{6} - \frac{181}{6272} a^{4} + \frac{219}{896} a^{2} - \frac{4485}{12544}$, $\frac{1}{25088} a^{17} - \frac{1}{25088} a^{16} - \frac{1}{12544} a^{15} + \frac{1}{12544} a^{14} + \frac{5}{12544} a^{13} - \frac{5}{12544} a^{12} + \frac{9}{1792} a^{11} - \frac{9}{1792} a^{10} + \frac{25}{6272} a^{9} - \frac{25}{6272} a^{8} + \frac{533}{12544} a^{7} - \frac{533}{12544} a^{6} + \frac{603}{12544} a^{5} - \frac{603}{12544} a^{4} + \frac{107}{1792} a^{3} - \frac{107}{1792} a^{2} - \frac{6053}{25088} a + \frac{6053}{25088}$, $\frac{1}{5056430569631971160403299020898636288} a^{18} - \frac{13643581151720912187594817852703}{5056430569631971160403299020898636288} a^{16} - \frac{166487295045168306376678127764555}{316026910601998197525206188806164768} a^{14} - \frac{17139085514255011034822619220033}{90293403029142342150058911087475648} a^{12} + \frac{37895518678504277539248744588339101}{2528215284815985580201649510449318144} a^{10} + \frac{37422550774808510163242110844032729}{2528215284815985580201649510449318144} a^{8} + \frac{31030808506790435488474017227497107}{632053821203996395050412377612329536} a^{6} - \frac{1367328482548010860586369458971385}{11286675378642792768757363885934456} a^{4} + \frac{529202002860392940087057014382023645}{5056430569631971160403299020898636288} a^{2} + \frac{264300242901835736507921740280151859}{722347224233138737200471288699805184}$, $\frac{1}{71079832236515483040733540972664692472210432} a^{19} - \frac{1}{10112861139263942320806598041797272576} a^{18} - \frac{373371297080069481363383468772876753223}{71079832236515483040733540972664692472210432} a^{17} + \frac{13643581151720912187594817852703}{10112861139263942320806598041797272576} a^{16} + \frac{74896238406790986701261595959467885203}{2538565437018410108597626463309453302578944} a^{15} - \frac{744885242149675870587957974683587}{2528215284815985580201649510449318144} a^{14} + \frac{8182950462392642418114926924562617767489}{17769958059128870760183385243166173118052608} a^{13} + \frac{235825945647131321511741022117467}{361173612116569368600235644349902592} a^{12} + \frac{60145870379308499656036913647893001860831}{35539916118257741520366770486332346236105216} a^{11} + \frac{72149566263262951956135553299521845}{5056430569631971160403299020898636288} a^{10} + \frac{465232028312954765393986501518278880399871}{35539916118257741520366770486332346236105216} a^{9} + \frac{100839222613565701254035596758664357}{5056430569631971160403299020898636288} a^{8} - \frac{584117626148149986060547067927577059618335}{17769958059128870760183385243166173118052608} a^{7} + \frac{80432659641784387728613684604928293}{2528215284815985580201649510449318144} a^{6} - \frac{745464220852512069218171460424839061944715}{17769958059128870760183385243166173118052608} a^{5} + \frac{36993338817164771227539095119347235}{361173612116569368600235644349902592} a^{4} + \frac{14288472128799153161438714113774376773024937}{71079832236515483040733540972664692472210432} a^{3} - \frac{2397146778025775143316400737504176113}{10112861139263942320806598041797272576} a^{2} + \frac{17415097750358552959410115626135030739948145}{71079832236515483040733540972664692472210432} a - \frac{107092978699311122943087029011779079}{1444694448466277474400942577399610368}$

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

Class group and class number

$C_{2}\times C_{122604504}$, which has order $245209008$ (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:  \( -1 \) (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:  \( 19344397.966990974 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-170}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{10}, \sqrt{-17})\), 5.5.390625.1, 10.10.25000000000000000.1, 10.0.35496425000000000000000.1, 10.0.221852656250000000000.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
17Data not computed