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Properties

Label 20.0.44574815118...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{35}\cdot 11^{10}$
Root discriminant $96.04$
Ramified primes $3, 5, 11$
Class number $1366888$ (GRH)
Class group $[2, 2, 341722]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43704064801, -33981478720, 30122086400, -16076932840, 18016281600, -4542160121, 5901800640, -665395200, 1137644880, -50828800, 131366161, -2079360, 9318400, -43320, 409600, -361, 10880, 0, 160, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 160*x^18 + 10880*x^16 - 361*x^15 + 409600*x^14 - 43320*x^13 + 9318400*x^12 - 2079360*x^11 + 131366161*x^10 - 50828800*x^9 + 1137644880*x^8 - 665395200*x^7 + 5901800640*x^6 - 4542160121*x^5 + 18016281600*x^4 - 16076932840*x^3 + 30122086400*x^2 - 33981478720*x + 43704064801)
 
gp: K = bnfinit(x^20 + 160*x^18 + 10880*x^16 - 361*x^15 + 409600*x^14 - 43320*x^13 + 9318400*x^12 - 2079360*x^11 + 131366161*x^10 - 50828800*x^9 + 1137644880*x^8 - 665395200*x^7 + 5901800640*x^6 - 4542160121*x^5 + 18016281600*x^4 - 16076932840*x^3 + 30122086400*x^2 - 33981478720*x + 43704064801, 1)
 

Normalized defining polynomial

\( x^{20} + 160 x^{18} + 10880 x^{16} - 361 x^{15} + 409600 x^{14} - 43320 x^{13} + 9318400 x^{12} - 2079360 x^{11} + 131366161 x^{10} - 50828800 x^{9} + 1137644880 x^{8} - 665395200 x^{7} + 5901800640 x^{6} - 4542160121 x^{5} + 18016281600 x^{4} - 16076932840 x^{3} + 30122086400 x^{2} - 33981478720 x + 43704064801 \)

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:  \(4457481511823277105577290058135986328125=3^{10}\cdot 5^{35}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.04$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(825=3\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{825}(1,·)$, $\chi_{825}(197,·)$, $\chi_{825}(263,·)$, $\chi_{825}(428,·)$, $\chi_{825}(331,·)$, $\chi_{825}(98,·)$, $\chi_{825}(527,·)$, $\chi_{825}(529,·)$, $\chi_{825}(661,·)$, $\chi_{825}(32,·)$, $\chi_{825}(34,·)$, $\chi_{825}(166,·)$, $\chi_{825}(593,·)$, $\chi_{825}(362,·)$, $\chi_{825}(199,·)$, $\chi_{825}(364,·)$, $\chi_{825}(694,·)$, $\chi_{825}(496,·)$, $\chi_{825}(692,·)$, $\chi_{825}(758,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{89} a^{10} - \frac{9}{89} a^{8} + \frac{15}{89} a^{6} + \frac{42}{89} a^{5} - \frac{32}{89} a^{4} - \frac{11}{89} a^{3} - \frac{39}{89} a^{2} + \frac{1}{89} a + \frac{16}{89}$, $\frac{1}{89} a^{11} - \frac{9}{89} a^{9} + \frac{15}{89} a^{7} + \frac{42}{89} a^{6} - \frac{32}{89} a^{5} - \frac{11}{89} a^{4} - \frac{39}{89} a^{3} + \frac{1}{89} a^{2} + \frac{16}{89} a$, $\frac{1}{89} a^{12} + \frac{23}{89} a^{8} + \frac{42}{89} a^{7} + \frac{14}{89} a^{6} + \frac{11}{89} a^{5} + \frac{29}{89} a^{4} - \frac{9}{89} a^{3} + \frac{21}{89} a^{2} + \frac{9}{89} a - \frac{34}{89}$, $\frac{1}{2747285859961} a^{13} + \frac{3742310140}{2747285859961} a^{12} + \frac{104}{2747285859961} a^{11} - \frac{11158791948}{2747285859961} a^{10} + \frac{4160}{2747285859961} a^{9} + \frac{833018707832}{2747285859961} a^{8} + \frac{1203866917383}{2747285859961} a^{7} - \frac{1263551811697}{2747285859961} a^{6} - \frac{339551439467}{2747285859961} a^{5} + \frac{416677346177}{2747285859961} a^{4} + \frac{432160308174}{2747285859961} a^{3} + \frac{245057541026}{2747285859961} a^{2} + \frac{1265607006281}{2747285859961} a + \frac{54037341880}{2747285859961}$, $\frac{1}{2747285859961} a^{14} + \frac{112}{2747285859961} a^{12} + \frac{929899329}{2747285859961} a^{11} + \frac{4928}{2747285859961} a^{10} + \frac{884409032626}{2747285859961} a^{9} + \frac{107520}{2747285859961} a^{8} - \frac{36084208150}{2747285859961} a^{7} - \frac{92603937123}{2747285859961} a^{6} + \frac{420872499370}{2747285859961} a^{5} - \frac{956913371391}{2747285859961} a^{4} + \frac{1218757947717}{2747285859961} a^{3} + \frac{586512073587}{2747285859961} a^{2} - \frac{420678902136}{2747285859961} a + \frac{4194304}{2747285859961}$, $\frac{1}{2747285859961} a^{15} + \frac{13948489935}{2747285859961} a^{12} - \frac{6720}{2747285859961} a^{11} + \frac{4275479821}{2747285859961} a^{10} - \frac{358400}{2747285859961} a^{9} - \frac{1037721942824}{2747285859961} a^{8} + \frac{1358200998316}{2747285859961} a^{7} + \frac{654098094361}{2747285859961} a^{6} - \frac{926128483806}{2747285859961} a^{5} - \frac{473874509914}{2747285859961} a^{4} - \frac{802899018074}{2747285859961} a^{3} + \frac{1087417364114}{2747285859961} a^{2} + \frac{123096034436}{2747285859961} a + \frac{121493799240}{2747285859961}$, $\frac{1}{2747285859961} a^{16} - \frac{7680}{2747285859961} a^{12} + \frac{4446407684}{2747285859961} a^{11} - \frac{450560}{2747285859961} a^{10} - \frac{691332643365}{2747285859961} a^{9} - \frac{11059200}{2747285859961} a^{8} + \frac{507555269104}{2747285859961} a^{7} + \frac{555498727506}{2747285859961} a^{6} - \frac{390793664543}{2747285859961} a^{5} + \frac{246213040392}{2747285859961} a^{4} - \frac{741045885832}{2747285859961} a^{3} - \frac{773219460665}{2747285859961} a^{2} - \frac{111665890837}{2747285859961} a - \frac{503316480}{2747285859961}$, $\frac{1}{2747285859961} a^{17} + \frac{6926084865}{2747285859961} a^{12} + \frac{348160}{2747285859961} a^{11} + \frac{9742072746}{2747285859961} a^{10} + \frac{20889600}{2747285859961} a^{9} + \frac{650251744063}{2747285859961} a^{8} + \frac{679585666262}{2747285859961} a^{7} - \frac{252472223577}{2747285859961} a^{6} - \frac{859323430812}{2747285859961} a^{5} - \frac{537081350710}{2747285859961} a^{4} + \frac{52259330849}{2747285859961} a^{3} - \frac{516195732524}{2747285859961} a^{2} + \frac{797378651705}{2747285859961} a - \frac{234668161548}{2747285859961}$, $\frac{1}{2747285859961} a^{18} + \frac{417792}{2747285859961} a^{12} - \frac{598002887}{2747285859961} a^{11} + \frac{27574272}{2747285859961} a^{10} + \frac{1162700132213}{2747285859961} a^{9} + \frac{721944576}{2747285859961} a^{8} + \frac{542101357346}{2747285859961} a^{7} - \frac{762725312057}{2747285859961} a^{6} - \frac{1187998019434}{2747285859961} a^{5} - \frac{596897708469}{2747285859961} a^{4} + \frac{1294315996787}{2747285859961} a^{3} + \frac{416494194506}{2747285859961} a^{2} + \frac{61761109022}{2747285859961} a + \frac{36507222016}{2747285859961}$, $\frac{1}{2747285859961} a^{19} + \frac{6502108532}{2747285859961} a^{12} - \frac{15876096}{2747285859961} a^{11} - \frac{7791998503}{2747285859961} a^{10} - \frac{1016070144}{2747285859961} a^{9} + \frac{1249696714782}{2747285859961} a^{8} + \frac{99087838340}{2747285859961} a^{7} + \frac{535482374536}{2747285859961} a^{6} - \frac{1093560228987}{2747285859961} a^{5} + \frac{916146978043}{2747285859961} a^{4} - \frac{211947147810}{2747285859961} a^{3} - \frac{749722331106}{2747285859961} a^{2} + \frac{372223248985}{2747285859961} a + \frac{517374626048}{2747285859961}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{341722}$, which has order $1366888$ (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:  \( 161406.837641 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.136125.2, 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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 $20$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
3Data not computed
5Data not computed
$11$11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$