↑

Properties

Label 20.0.16000000000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{26}$
Root discriminant $22.92$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![225, 0, 250, 0, 575, 0, -500, 0, 725, 0, -390, 0, 525, 0, -160, 0, 55, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 + 55*x^16 - 160*x^14 + 525*x^12 - 390*x^10 + 725*x^8 - 500*x^6 + 575*x^4 + 250*x^2 + 225)
 
gp: K = bnfinit(x^20 - 10*x^18 + 55*x^16 - 160*x^14 + 525*x^12 - 390*x^10 + 725*x^8 - 500*x^6 + 575*x^4 + 250*x^2 + 225, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} + 55 x^{16} - 160 x^{14} + 525 x^{12} - 390 x^{10} + 725 x^{8} - 500 x^{6} + 575 x^{4} + 250 x^{2} + 225 \)

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:  \(1600000000000000000000000000=2^{30}\cdot 5^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.92$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{120} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{7}{24} a^{2} - \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{120} a^{11} - \frac{1}{24} a^{9} + \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{11}{24} a^{3} - \frac{1}{2} a^{2} + \frac{1}{24} a$, $\frac{1}{120} a^{12} - \frac{1}{4} a^{7} + \frac{1}{6} a^{4} - \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{120} a^{13} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{360} a^{14} - \frac{1}{360} a^{10} - \frac{1}{24} a^{8} + \frac{7}{72} a^{6} - \frac{5}{24} a^{4} + \frac{1}{36} a^{2} - \frac{3}{8}$, $\frac{1}{720} a^{15} - \frac{1}{720} a^{14} - \frac{1}{720} a^{11} + \frac{1}{720} a^{10} - \frac{1}{48} a^{9} + \frac{1}{48} a^{8} - \frac{29}{144} a^{7} + \frac{29}{144} a^{6} - \frac{5}{48} a^{5} - \frac{19}{48} a^{4} - \frac{35}{72} a^{3} + \frac{35}{72} a^{2} + \frac{1}{16} a + \frac{7}{16}$, $\frac{1}{2160} a^{16} - \frac{1}{2160} a^{14} - \frac{1}{2160} a^{12} + \frac{1}{540} a^{10} - \frac{1}{54} a^{8} - \frac{1}{4} a^{7} - \frac{5}{54} a^{6} - \frac{1}{432} a^{4} + \frac{61}{432} a^{2} - \frac{1}{4} a - \frac{11}{48}$, $\frac{1}{2160} a^{17} - \frac{1}{2160} a^{15} - \frac{1}{2160} a^{13} + \frac{1}{540} a^{11} - \frac{1}{54} a^{9} + \frac{17}{108} a^{7} - \frac{1}{4} a^{6} - \frac{1}{432} a^{5} - \frac{155}{432} a^{3} - \frac{1}{2} a^{2} - \frac{23}{48} a - \frac{1}{4}$, $\frac{1}{1810080} a^{18} - \frac{1}{4320} a^{17} - \frac{1}{18855} a^{16} + \frac{1}{4320} a^{15} + \frac{1013}{905040} a^{14} - \frac{17}{4320} a^{13} - \frac{81}{67040} a^{12} - \frac{1}{1080} a^{11} + \frac{1057}{301680} a^{10} + \frac{1}{108} a^{9} - \frac{271}{6704} a^{8} - \frac{11}{54} a^{7} - \frac{3359}{362016} a^{6} - \frac{71}{864} a^{5} + \frac{12095}{60336} a^{4} + \frac{371}{864} a^{3} - \frac{4079}{22626} a^{2} + \frac{29}{96} a + \frac{17905}{40224}$, $\frac{1}{1810080} a^{19} + \frac{323}{1810080} a^{17} - \frac{1}{4320} a^{16} - \frac{907}{1810080} a^{15} + \frac{1}{4320} a^{14} + \frac{617}{226260} a^{13} - \frac{17}{4320} a^{12} - \frac{569}{226260} a^{11} - \frac{1}{1080} a^{10} + \frac{145}{11313} a^{9} + \frac{1}{108} a^{8} - \frac{82969}{362016} a^{7} - \frac{11}{54} a^{6} - \frac{86231}{362016} a^{5} - \frac{71}{864} a^{4} - \frac{19939}{120672} a^{3} - \frac{61}{864} a^{2} - \frac{4637}{10056} a + \frac{29}{96}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1313342.83504 \) (assuming GRH)
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

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{5})\), 5.1.1000000.1 x5, 10.0.40000000000000.4, 10.2.5000000000000.2 x5, 10.0.8000000000000.2 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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
5Data not computed