↑

Properties

Label 20.0.55909223440...1849.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 79^{10}$
Root discriminant $15.39$
Ramified primes $3, 79$
Class number $1$
Class group Trivial
Galois group $D_{10}$ (as 20T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 4, -55, 297, -333, 238, -571, 457, -70, -75, 161, 37, -82, 88, -15, -6, 8, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 + 8*x^17 - 6*x^16 - 15*x^15 + 88*x^14 - 82*x^13 + 37*x^12 + 161*x^11 - 75*x^10 - 70*x^9 + 457*x^8 - 571*x^7 + 238*x^6 - 333*x^5 + 297*x^4 - 55*x^3 + 4*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + x^18 + 8*x^17 - 6*x^16 - 15*x^15 + 88*x^14 - 82*x^13 + 37*x^12 + 161*x^11 - 75*x^10 - 70*x^9 + 457*x^8 - 571*x^7 + 238*x^6 - 333*x^5 + 297*x^4 - 55*x^3 + 4*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + x^{18} + 8 x^{17} - 6 x^{16} - 15 x^{15} + 88 x^{14} - 82 x^{13} + 37 x^{12} + 161 x^{11} - 75 x^{10} - 70 x^{9} + 457 x^{8} - 571 x^{7} + 238 x^{6} - 333 x^{5} + 297 x^{4} - 55 x^{3} + 4 x^{2} - 5 x + 1 \)

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:  \(559092234403032700371849=3^{10}\cdot 79^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.39$
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:  $3, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{14} - \frac{4}{9} a^{13} - \frac{1}{3} a^{12} + \frac{4}{9} a^{11} + \frac{2}{9} a^{10} - \frac{2}{9} a^{9} - \frac{2}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{133713} a^{18} + \frac{3023}{133713} a^{17} + \frac{2936}{44571} a^{16} - \frac{11183}{133713} a^{15} + \frac{15670}{133713} a^{14} - \frac{62468}{133713} a^{13} + \frac{66340}{133713} a^{12} + \frac{44401}{133713} a^{11} + \frac{2183}{133713} a^{10} + \frac{13669}{44571} a^{9} + \frac{65983}{133713} a^{8} - \frac{1345}{133713} a^{7} + \frac{29873}{133713} a^{6} - \frac{2623}{133713} a^{5} - \frac{33778}{133713} a^{4} - \frac{55027}{133713} a^{3} - \frac{9656}{44571} a^{2} - \frac{12956}{133713} a + \frac{54695}{133713}$, $\frac{1}{1224552528805520511} a^{19} - \frac{1032491912798}{408184176268506837} a^{18} - \frac{1385560461684462}{136061392089502279} a^{17} + \frac{16367728475116}{441598459720707} a^{16} - \frac{44333360843718520}{1224552528805520511} a^{15} + \frac{14684857875879857}{136061392089502279} a^{14} + \frac{502337802237598312}{1224552528805520511} a^{13} - \frac{318538490245761307}{1224552528805520511} a^{12} + \frac{447471073293651667}{1224552528805520511} a^{11} - \frac{351604643425215683}{1224552528805520511} a^{10} + \frac{1290478606537511}{1224552528805520511} a^{9} - \frac{348701208910477763}{1224552528805520511} a^{8} + \frac{43198732004852}{187154597096977} a^{7} - \frac{548133728075167267}{1224552528805520511} a^{6} + \frac{440664239840108386}{1224552528805520511} a^{5} - \frac{131050927925684348}{408184176268506837} a^{4} + \frac{3360219307804210}{136061392089502279} a^{3} - \frac{94203431654066276}{1224552528805520511} a^{2} + \frac{8032157109832}{2280358526639703} a - \frac{129521375586828328}{408184176268506837}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{96092166810790}{27474199116141} a^{19} - \frac{167108841371303}{27474199116141} a^{18} + \frac{51753547767158}{27474199116141} a^{17} + \frac{282461861815}{9907753017} a^{16} - \frac{124121208803925}{9158066372047} a^{15} - \frac{1544404900966796}{27474199116141} a^{14} + \frac{8053937687745626}{27474199116141} a^{13} - \frac{1922343285258563}{9158066372047} a^{12} + \frac{1992656430872606}{27474199116141} a^{11} + \frac{5339033228129410}{9158066372047} a^{10} - \frac{3044823086315212}{27474199116141} a^{9} - \frac{7646021619585541}{27474199116141} a^{8} + \frac{57624992305718}{37791195483} a^{7} - \frac{43912315241752433}{27474199116141} a^{6} + \frac{3696876718234588}{9158066372047} a^{5} - \frac{28889125995667657}{27474199116141} a^{4} + \frac{20925564741740648}{27474199116141} a^{3} + \frac{401789753313739}{27474199116141} a^{2} + \frac{380716560077269}{27474199116141} a - \frac{353841031305115}{27474199116141} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12895.8301109 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-79}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{237}) \), \(\Q(\sqrt{-3}, \sqrt{-79})\), 5.1.6241.1 x5, 10.0.3077056399.1, 10.0.9464869683.1 x5, 10.2.747724704957.1 x5

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$79$79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$