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Properties

Label 20.0.72825983445...2832.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 3^{10}\cdot 7^{5}\cdot 19^{10}\cdot 43^{8}$
Root discriminant $78.18$
Ramified primes $2, 3, 7, 19, 43$
Class number $3232$ (GRH)
Class group $[2, 4, 404]$ (GRH)
Galois group $D_4\times D_5$ (as 20T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1823113, 1084745, 1041518, 87755, 872126, 574249, 365450, -258674, 523033, -238292, 163342, -48534, 27597, -8472, 5219, -1892, 822, -221, 61, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 61*x^18 - 221*x^17 + 822*x^16 - 1892*x^15 + 5219*x^14 - 8472*x^13 + 27597*x^12 - 48534*x^11 + 163342*x^10 - 238292*x^9 + 523033*x^8 - 258674*x^7 + 365450*x^6 + 574249*x^5 + 872126*x^4 + 87755*x^3 + 1041518*x^2 + 1084745*x + 1823113)
 
gp: K = bnfinit(x^20 - 9*x^19 + 61*x^18 - 221*x^17 + 822*x^16 - 1892*x^15 + 5219*x^14 - 8472*x^13 + 27597*x^12 - 48534*x^11 + 163342*x^10 - 238292*x^9 + 523033*x^8 - 258674*x^7 + 365450*x^6 + 574249*x^5 + 872126*x^4 + 87755*x^3 + 1041518*x^2 + 1084745*x + 1823113, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 61 x^{18} - 221 x^{17} + 822 x^{16} - 1892 x^{15} + 5219 x^{14} - 8472 x^{13} + 27597 x^{12} - 48534 x^{11} + 163342 x^{10} - 238292 x^{9} + 523033 x^{8} - 258674 x^{7} + 365450 x^{6} + 574249 x^{5} + 872126 x^{4} + 87755 x^{3} + 1041518 x^{2} + 1084745 x + 1823113 \)

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:  \(72825983445554038731855441697381112832=2^{10}\cdot 3^{10}\cdot 7^{5}\cdot 19^{10}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.18$
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:  $2, 3, 7, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{104253} a^{18} + \frac{12872}{104253} a^{17} + \frac{2137}{104253} a^{16} + \frac{2366}{104253} a^{15} + \frac{8462}{104253} a^{14} - \frac{1228}{3363} a^{13} + \frac{471}{1121} a^{12} + \frac{40291}{104253} a^{11} - \frac{14967}{34751} a^{10} - \frac{9502}{104253} a^{9} - \frac{6301}{34751} a^{8} - \frac{16706}{104253} a^{7} - \frac{37415}{104253} a^{6} - \frac{3774}{34751} a^{5} - \frac{17800}{104253} a^{4} + \frac{23578}{104253} a^{3} - \frac{4301}{104253} a^{2} + \frac{9620}{104253} a + \frac{10892}{104253}$, $\frac{1}{341179666298691837497583326152809271091305698920386719872049} a^{19} + \frac{1317986162838489280502340143188020355773395606467458473}{341179666298691837497583326152809271091305698920386719872049} a^{18} + \frac{156815530682175076260479574136335272269284987236460086213}{341179666298691837497583326152809271091305698920386719872049} a^{17} + \frac{3450014404057745663615526530552205594682387209430112518}{1572256526722082200449692747247968991204173727743717603097} a^{16} + \frac{37639653811785538791863858099695661710644501525967857048334}{341179666298691837497583326152809271091305698920386719872049} a^{15} + \frac{33144430258406635364835626762791396221484765602540018975514}{341179666298691837497583326152809271091305698920386719872049} a^{14} + \frac{27181210387845626714412597847376717575867650476898233724}{62179636650025849735298583224495948804684836690429509727} a^{13} + \frac{85168217893384167833443490358446243339590283248637417216945}{341179666298691837497583326152809271091305698920386719872049} a^{12} + \frac{735686433436897351938541324728058827489768840256980808145}{2145784064771646776714360541841567742712614458618784401711} a^{11} - \frac{36283027560878250853763856065407474727991261082148621580431}{113726555432897279165861108717603090363768566306795573290683} a^{10} - \frac{24013840893932834369444799003687427650167506039332223317859}{48739952328384548213940475164687038727329385560055245696007} a^{9} - \frac{30594369790052127567160838359362881899445287266934879373152}{341179666298691837497583326152809271091305698920386719872049} a^{8} + \frac{48346034559546301890185678309950109243690289547713938708711}{341179666298691837497583326152809271091305698920386719872049} a^{7} + \frac{75363388884111664476884205782514145509007973241956314578703}{341179666298691837497583326152809271091305698920386719872049} a^{6} + \frac{932238574749306690930944435341737531746626014295515002020}{2565260648862344642838972377088791511964704503160802405053} a^{5} - \frac{13311503768654960249258317367502741201582809936260454477114}{113726555432897279165861108717603090363768566306795573290683} a^{4} - \frac{96556956274626590270308747137814713394956171030972762889}{275366962307257334542036582851339201849318562486187828791} a^{3} - \frac{108254721599578380577115204364309721385788153083378010762984}{341179666298691837497583326152809271091305698920386719872049} a^{2} + \frac{135069308047104826939476917744131627593275710334468492767249}{341179666298691837497583326152809271091305698920386719872049} a - \frac{72699698413065108551251955085196279984758837776378240779112}{341179666298691837497583326152809271091305698920386719872049}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{404}$, which has order $3232$ (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:  \( 13701218.918 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times D_5$ (as 20T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $D_4\times D_5$
Character table for $D_4\times D_5$

Intermediate fields

\(\Q(\sqrt{57}) \), 4.0.90972.2, 5.5.667489.1, 10.10.2057065406163657.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ R $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ R $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$