↑

Properties

Label 20.0.45582459238...3584.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{18}\cdot 1429^{8}$
Root discriminant $34.11$
Ramified primes $2, 1429$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2\times C_2^4:D_5$ (as 20T85)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, 0, -26244, 0, 7290, 0, -3402, 0, 1629, 0, -580, 0, 181, 0, -42, 0, 10, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 + 10*x^16 - 42*x^14 + 181*x^12 - 580*x^10 + 1629*x^8 - 3402*x^6 + 7290*x^4 - 26244*x^2 + 59049)
 
gp: K = bnfinit(x^20 - 4*x^18 + 10*x^16 - 42*x^14 + 181*x^12 - 580*x^10 + 1629*x^8 - 3402*x^6 + 7290*x^4 - 26244*x^2 + 59049, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{18} + 10 x^{16} - 42 x^{14} + 181 x^{12} - 580 x^{10} + 1629 x^{8} - 3402 x^{6} + 7290 x^{4} - 26244 x^{2} + 59049 \)

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:  \(4558245923829641289021197123584=2^{18}\cdot 1429^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.11$
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:  $2, 1429$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{1}{8} a^{8} - \frac{1}{12} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{11}{24} a^{3} - \frac{3}{8} a^{2} + \frac{1}{12} a - \frac{3}{8}$, $\frac{1}{72} a^{12} - \frac{1}{18} a^{10} - \frac{1}{9} a^{8} + \frac{1}{6} a^{6} - \frac{1}{9} a^{4} + \frac{4}{9} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{216} a^{13} - \frac{1}{54} a^{11} + \frac{5}{108} a^{9} + \frac{1}{18} a^{7} - \frac{1}{4} a^{6} - \frac{1}{27} a^{5} - \frac{47}{108} a^{3} - \frac{1}{2} a^{2} - \frac{5}{24} a - \frac{1}{4}$, $\frac{1}{1296} a^{14} + \frac{5}{1296} a^{12} - \frac{13}{648} a^{10} - \frac{1}{8} a^{9} - \frac{19}{216} a^{8} - \frac{1}{8} a^{7} + \frac{13}{81} a^{6} - \frac{83}{648} a^{4} + \frac{1}{8} a^{3} - \frac{13}{48} a^{2} + \frac{1}{8} a - \frac{5}{16}$, $\frac{1}{7776} a^{15} - \frac{1}{2592} a^{14} - \frac{13}{7776} a^{13} + \frac{13}{2592} a^{12} - \frac{29}{1944} a^{11} + \frac{29}{648} a^{10} + \frac{5}{1296} a^{9} - \frac{5}{432} a^{8} + \frac{77}{3888} a^{7} - \frac{77}{1296} a^{6} - \frac{127}{1944} a^{5} + \frac{127}{648} a^{4} + \frac{41}{864} a^{3} + \frac{103}{288} a^{2} - \frac{29}{96} a + \frac{13}{32}$, $\frac{1}{93312} a^{16} + \frac{7}{46656} a^{14} + \frac{181}{93312} a^{12} + \frac{347}{15552} a^{10} - \frac{83}{23328} a^{8} + \frac{5713}{46656} a^{6} - \frac{2491}{10368} a^{4} - \frac{1}{2} a^{3} - \frac{17}{576} a^{2} - \frac{63}{128}$, $\frac{1}{559872} a^{17} - \frac{1}{186624} a^{16} + \frac{7}{279936} a^{15} - \frac{7}{93312} a^{14} - \frac{1115}{559872} a^{13} + \frac{1115}{186624} a^{12} + \frac{1211}{93312} a^{11} - \frac{1211}{31104} a^{10} - \frac{14987}{139968} a^{9} + \frac{3323}{46656} a^{8} + \frac{9601}{279936} a^{7} + \frac{13727}{93312} a^{6} + \frac{3845}{62208} a^{5} - \frac{3845}{20736} a^{4} - \frac{331}{1152} a^{3} + \frac{43}{384} a^{2} - \frac{79}{768} a + \frac{15}{256}$, $\frac{1}{3359232} a^{18} + \frac{5}{3359232} a^{16} - \frac{1241}{3359232} a^{14} + \frac{5767}{1119744} a^{12} + \frac{77297}{1679616} a^{10} - \frac{569}{1679616} a^{8} + \frac{5249}{124416} a^{6} + \frac{4537}{41472} a^{4} + \frac{371}{4608} a^{2} - \frac{1}{2} a + \frac{207}{512}$, $\frac{1}{20155392} a^{19} - \frac{1}{6718464} a^{18} + \frac{5}{20155392} a^{17} - \frac{5}{6718464} a^{16} - \frac{1241}{20155392} a^{15} + \frac{1241}{6718464} a^{14} - \frac{9785}{6718464} a^{13} + \frac{9785}{2239488} a^{12} + \frac{170609}{10077696} a^{11} - \frac{170609}{3359232} a^{10} - \frac{233849}{10077696} a^{9} - \frac{186055}{3359232} a^{8} + \frac{77825}{746496} a^{7} + \frac{15487}{248832} a^{6} + \frac{50617}{248832} a^{5} - \frac{9145}{82944} a^{4} + \frac{2897}{9216} a^{3} + \frac{559}{3072} a^{2} + \frac{655}{3072} a + \frac{497}{1024}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 14385341.0336 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T85):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.2042041.1, 10.0.2135004900188672.1, 10.0.33359451565448.1, 10.10.266875612523584.1

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
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
1429Data not computed