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Properties

Label 20.0.19457712071...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 41^{16}$
Root discriminant $65.23$
Ramified primes $5, 41$
Class number $505$ (GRH)
Class group $[505]$ (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![6561, 15309, 39366, 88695, 202338, 82674, 153711, 59106, 108670, 44804, 73952, 27901, 45177, 6496, 3849, 679, 294, -38, 17, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 17*x^18 - 38*x^17 + 294*x^16 + 679*x^15 + 3849*x^14 + 6496*x^13 + 45177*x^12 + 27901*x^11 + 73952*x^10 + 44804*x^9 + 108670*x^8 + 59106*x^7 + 153711*x^6 + 82674*x^5 + 202338*x^4 + 88695*x^3 + 39366*x^2 + 15309*x + 6561)
 
gp: K = bnfinit(x^20 - x^19 + 17*x^18 - 38*x^17 + 294*x^16 + 679*x^15 + 3849*x^14 + 6496*x^13 + 45177*x^12 + 27901*x^11 + 73952*x^10 + 44804*x^9 + 108670*x^8 + 59106*x^7 + 153711*x^6 + 82674*x^5 + 202338*x^4 + 88695*x^3 + 39366*x^2 + 15309*x + 6561, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 17 x^{18} - 38 x^{17} + 294 x^{16} + 679 x^{15} + 3849 x^{14} + 6496 x^{13} + 45177 x^{12} + 27901 x^{11} + 73952 x^{10} + 44804 x^{9} + 108670 x^{8} + 59106 x^{7} + 153711 x^{6} + 82674 x^{5} + 202338 x^{4} + 88695 x^{3} + 39366 x^{2} + 15309 x + 6561 \)

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:  \(1945771207112214793287128936767578125=5^{15}\cdot 41^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.23$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(205=5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{205}(1,·)$, $\chi_{205}(133,·)$, $\chi_{205}(201,·)$, $\chi_{205}(139,·)$, $\chi_{205}(141,·)$, $\chi_{205}(78,·)$, $\chi_{205}(16,·)$, $\chi_{205}(18,·)$, $\chi_{205}(83,·)$, $\chi_{205}(92,·)$, $\chi_{205}(98,·)$, $\chi_{205}(37,·)$, $\chi_{205}(42,·)$, $\chi_{205}(174,·)$, $\chi_{205}(51,·)$, $\chi_{205}(182,·)$, $\chi_{205}(119,·)$, $\chi_{205}(57,·)$, $\chi_{205}(59,·)$, $\chi_{205}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{4}{9} a^{7} - \frac{2}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{15} - \frac{1}{27} a^{14} - \frac{1}{27} a^{13} - \frac{2}{27} a^{12} - \frac{1}{9} a^{11} + \frac{4}{27} a^{10} - \frac{1}{9} a^{9} - \frac{11}{27} a^{8} - \frac{4}{9} a^{7} + \frac{10}{27} a^{6} + \frac{8}{27} a^{5} + \frac{11}{27} a^{4} - \frac{5}{27} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{243} a^{16} + \frac{2}{243} a^{15} - \frac{13}{243} a^{14} + \frac{40}{243} a^{13} - \frac{4}{27} a^{12} - \frac{5}{243} a^{11} - \frac{1}{9} a^{10} + \frac{34}{243} a^{9} - \frac{2}{9} a^{8} + \frac{28}{243} a^{7} + \frac{110}{243} a^{6} - \frac{100}{243} a^{5} - \frac{53}{243} a^{4} - \frac{4}{81} a^{3} + \frac{4}{27} a^{2} - \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{1940034077006771439} a^{17} + \frac{2935784494971791}{1940034077006771439} a^{16} + \frac{7068803377354901}{1940034077006771439} a^{15} + \frac{39903748542193750}{1940034077006771439} a^{14} + \frac{19506061006875230}{646678025668923813} a^{13} + \frac{99659815298340970}{1940034077006771439} a^{12} + \frac{89212581291387662}{646678025668923813} a^{11} + \frac{179032193262409957}{1940034077006771439} a^{10} + \frac{49470807667800659}{646678025668923813} a^{9} - \frac{188278524255186755}{1940034077006771439} a^{8} - \frac{245136774381446305}{1940034077006771439} a^{7} + \frac{447406356616225346}{1940034077006771439} a^{6} + \frac{441760151534200933}{1940034077006771439} a^{5} - \frac{116084098808408099}{646678025668923813} a^{4} + \frac{96956908175324261}{215559341889641271} a^{3} + \frac{13338366656050996}{71853113963213757} a^{2} - \frac{9318296638845271}{23951037987737919} a + \frac{950506816119959}{7983679329245973}$, $\frac{1}{5820102231020314317} a^{18} - \frac{1}{5820102231020314317} a^{17} - \frac{7660871475167647}{5820102231020314317} a^{16} - \frac{3480816641708270}{5820102231020314317} a^{15} - \frac{39697708986991366}{1940034077006771439} a^{14} + \frac{365095956663452623}{5820102231020314317} a^{13} - \frac{50756508948401608}{1940034077006771439} a^{12} + \frac{823361956095325282}{5820102231020314317} a^{11} + \frac{155901308665264949}{1940034077006771439} a^{10} + \frac{848513619160618942}{5820102231020314317} a^{9} + \frac{2111893888275328985}{5820102231020314317} a^{8} + \frac{2104982030747554148}{5820102231020314317} a^{7} - \frac{2624344635089301917}{5820102231020314317} a^{6} + \frac{332893314856611409}{1940034077006771439} a^{5} + \frac{321709150583583914}{646678025668923813} a^{4} - \frac{13483724389364675}{215559341889641271} a^{3} - \frac{31965735359382364}{71853113963213757} a^{2} - \frac{4129465576718317}{23951037987737919} a - \frac{203965088156943}{2661226443081991}$, $\frac{1}{17460306693060942951} a^{19} - \frac{1}{17460306693060942951} a^{18} - \frac{1}{17460306693060942951} a^{17} - \frac{30423400402114379}{17460306693060942951} a^{16} + \frac{97536684618034007}{5820102231020314317} a^{15} - \frac{515992921693959470}{17460306693060942951} a^{14} - \frac{156743815216716868}{5820102231020314317} a^{13} - \frac{1710657394930186877}{17460306693060942951} a^{12} + \frac{133217674824838772}{5820102231020314317} a^{11} - \frac{1655865867799745855}{17460306693060942951} a^{10} + \frac{1364717556662550551}{17460306693060942951} a^{9} + \frac{1834885718851503170}{17460306693060942951} a^{8} + \frac{6281393238314236003}{17460306693060942951} a^{7} - \frac{1895702599374222971}{5820102231020314317} a^{6} + \frac{787698960578587621}{1940034077006771439} a^{5} + \frac{1145692751293769}{646678025668923813} a^{4} - \frac{32870889881382332}{215559341889641271} a^{3} - \frac{575421866121556}{23951037987737919} a^{2} + \frac{1198224680533094}{23951037987737919} a + \frac{1183919319596685}{2661226443081991}$

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

Class group and class number

$C_{505}$, which has order $505$ (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:  \( -\frac{37742873463880}{5820102231020314317} a^{19} - \frac{120777195084416}{5820102231020314317} a^{18} - \frac{415335701549948}{5820102231020314317} a^{17} - \frac{1328549145928576}{5820102231020314317} a^{16} - \frac{1305903421850248}{1940034077006771439} a^{15} - \frac{74813923780102936}{5820102231020314317} a^{14} - \frac{77644639289893936}{1940034077006771439} a^{13} - \frac{809453903593472374}{5820102231020314317} a^{12} - \frac{824455327944994720}{1940034077006771439} a^{11} - \frac{7773205178483628208}{5820102231020314317} a^{10} - \frac{4144831780906988288}{5820102231020314317} a^{9} - \frac{11518392769430976728}{5820102231020314317} a^{8} - \frac{6084705844284598201}{5820102231020314317} a^{7} - \frac{67544646350959648}{23951037987737919} a^{6} - \frac{288453684735049288}{215559341889641271} a^{5} - \frac{289767136731592312}{71853113963213757} a^{4} - \frac{42385246899937240}{23951037987737919} a^{3} - \frac{139191462630118925}{23951037987737919} a^{2} - \frac{822794641512584}{2661226443081991} a - \frac{362331585253248}{2661226443081991} \) (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:  \( 41023218.2567 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.2825761.1, 10.10.24952891341003125.1

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$41$41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.1$x^{5} - 41$$5$$1$$4$$C_5$$[\ ]_{5}$