↑

Properties

Label 20.0.70638883973...9408.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{15}\cdot 71^{9}$
Root discriminant $43.90$
Ramified primes $2, 3, 71$
Class number $164$ (GRH)
Class group $[2, 82]$ (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8641, -137098, 1079706, -668326, 857558, -515444, 407778, -242326, 153485, -91110, 56352, -33000, 18493, -9430, 4398, -1808, 657, -194, 48, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 48*x^18 - 194*x^17 + 657*x^16 - 1808*x^15 + 4398*x^14 - 9430*x^13 + 18493*x^12 - 33000*x^11 + 56352*x^10 - 91110*x^9 + 153485*x^8 - 242326*x^7 + 407778*x^6 - 515444*x^5 + 857558*x^4 - 668326*x^3 + 1079706*x^2 - 137098*x + 8641)
 
gp: K = bnfinit(x^20 - 8*x^19 + 48*x^18 - 194*x^17 + 657*x^16 - 1808*x^15 + 4398*x^14 - 9430*x^13 + 18493*x^12 - 33000*x^11 + 56352*x^10 - 91110*x^9 + 153485*x^8 - 242326*x^7 + 407778*x^6 - 515444*x^5 + 857558*x^4 - 668326*x^3 + 1079706*x^2 - 137098*x + 8641, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 48 x^{18} - 194 x^{17} + 657 x^{16} - 1808 x^{15} + 4398 x^{14} - 9430 x^{13} + 18493 x^{12} - 33000 x^{11} + 56352 x^{10} - 91110 x^{9} + 153485 x^{8} - 242326 x^{7} + 407778 x^{6} - 515444 x^{5} + 857558 x^{4} - 668326 x^{3} + 1079706 x^{2} - 137098 x + 8641 \)

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:  \(706388839731582814105670315409408=2^{30}\cdot 3^{15}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.90$
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:  $2, 3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{73} a^{18} + \frac{3}{73} a^{17} - \frac{31}{73} a^{16} + \frac{5}{73} a^{15} + \frac{23}{73} a^{14} + \frac{2}{73} a^{13} + \frac{19}{73} a^{12} - \frac{28}{73} a^{11} - \frac{3}{73} a^{10} + \frac{33}{73} a^{9} - \frac{35}{73} a^{8} + \frac{1}{73} a^{7} + \frac{28}{73} a^{6} + \frac{11}{73} a^{5} - \frac{22}{73} a^{4} - \frac{5}{73} a^{3} + \frac{27}{73} a^{2} - \frac{30}{73} a - \frac{33}{73}$, $\frac{1}{4661993259163710335665520046835519628032429515574543} a^{19} + \frac{9001552123970987285231012605220426458619446841538}{4661993259163710335665520046835519628032429515574543} a^{18} - \frac{1626989940423444259579396290380797183410905703972405}{4661993259163710335665520046835519628032429515574543} a^{17} - \frac{602866701275825984703978397122281464826938800398646}{4661993259163710335665520046835519628032429515574543} a^{16} + \frac{643913255325149302414506840555589456187640169586861}{4661993259163710335665520046835519628032429515574543} a^{15} + \frac{799794994113740119822623252131501011304993986464427}{4661993259163710335665520046835519628032429515574543} a^{14} - \frac{1670533569966610488835647909980823439903248950528689}{4661993259163710335665520046835519628032429515574543} a^{13} - \frac{222184682076653128138265371480405282900393117360925}{4661993259163710335665520046835519628032429515574543} a^{12} - \frac{1642051858701948787094647239287473585995024228226433}{4661993259163710335665520046835519628032429515574543} a^{11} + \frac{521667392097149285541282642431134411233653169361485}{4661993259163710335665520046835519628032429515574543} a^{10} + \frac{127105088531788086634110900835097318141132195223964}{4661993259163710335665520046835519628032429515574543} a^{9} - \frac{2017191442040128267082158920442314750237053822896843}{4661993259163710335665520046835519628032429515574543} a^{8} + \frac{470306279304358896978295351853566394094545986624494}{4661993259163710335665520046835519628032429515574543} a^{7} + \frac{1370995755075737969802004437903237129778868343484086}{4661993259163710335665520046835519628032429515574543} a^{6} + \frac{1639009118537131297319652966318012147201349687815607}{4661993259163710335665520046835519628032429515574543} a^{5} - \frac{247762406112189555322829529715194388984819672422265}{4661993259163710335665520046835519628032429515574543} a^{4} + \frac{1824364701526253113831286648494220812033226690369851}{4661993259163710335665520046835519628032429515574543} a^{3} + \frac{1634407025381130127106867535409462695802162151391330}{4661993259163710335665520046835519628032429515574543} a^{2} - \frac{1443507158174001797459777668400035706242544485787875}{4661993259163710335665520046835519628032429515574543} a - \frac{2175346406076252391979648642924055526963976896841621}{4661993259163710335665520046835519628032429515574543}$

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

Class group and class number

$C_{2}\times C_{82}$, which has order $164$ (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:  \( 528205.092142 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.0.122688.1, 10.10.6323239406592.1

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 $20$ $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
3Data not computed
$71$71.10.0.1$x^{10} - x + 22$$1$$10$$0$$C_{10}$$[\ ]^{10}$
71.10.9.7$x^{10} + 568$$10$$1$$9$$C_{10}$$[\ ]_{10}$