↑

Properties

Label 20.0.38028917784...0625.3
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 7^{10}\cdot 13^{10}$
Root discriminant $21.33$
Ramified primes $5, 7, 13$
Class number $4$
Class group $[4]$
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![1355, -5575, 12949, -21221, 26886, -28225, 26369, -23886, 20781, -16150, 10834, -6246, 3286, -1750, 904, -391, 141, -50, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 19*x^18 - 50*x^17 + 141*x^16 - 391*x^15 + 904*x^14 - 1750*x^13 + 3286*x^12 - 6246*x^11 + 10834*x^10 - 16150*x^9 + 20781*x^8 - 23886*x^7 + 26369*x^6 - 28225*x^5 + 26886*x^4 - 21221*x^3 + 12949*x^2 - 5575*x + 1355)
 
gp: K = bnfinit(x^20 - 6*x^19 + 19*x^18 - 50*x^17 + 141*x^16 - 391*x^15 + 904*x^14 - 1750*x^13 + 3286*x^12 - 6246*x^11 + 10834*x^10 - 16150*x^9 + 20781*x^8 - 23886*x^7 + 26369*x^6 - 28225*x^5 + 26886*x^4 - 21221*x^3 + 12949*x^2 - 5575*x + 1355, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 19 x^{18} - 50 x^{17} + 141 x^{16} - 391 x^{15} + 904 x^{14} - 1750 x^{13} + 3286 x^{12} - 6246 x^{11} + 10834 x^{10} - 16150 x^{9} + 20781 x^{8} - 23886 x^{7} + 26369 x^{6} - 28225 x^{5} + 26886 x^{4} - 21221 x^{3} + 12949 x^{2} - 5575 x + 1355 \)

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:  \(380289177849714310556640625=5^{10}\cdot 7^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.33$
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:  $5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{6} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{15} - \frac{2}{25} a^{13} + \frac{2}{25} a^{12} + \frac{2}{25} a^{10} + \frac{9}{25} a^{9} + \frac{6}{25} a^{8} + \frac{7}{25} a^{7} - \frac{9}{25} a^{6} + \frac{8}{25} a^{5} + \frac{12}{25} a^{4} + \frac{7}{25} a^{3} - \frac{8}{25} a^{2} - \frac{2}{5}$, $\frac{1}{25} a^{16} - \frac{2}{25} a^{14} + \frac{2}{25} a^{13} + \frac{2}{25} a^{11} - \frac{1}{25} a^{10} + \frac{6}{25} a^{9} + \frac{7}{25} a^{8} - \frac{9}{25} a^{7} + \frac{3}{25} a^{6} + \frac{12}{25} a^{5} + \frac{7}{25} a^{4} - \frac{8}{25} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{25} a^{17} + \frac{2}{25} a^{14} + \frac{1}{25} a^{13} + \frac{1}{25} a^{12} - \frac{1}{25} a^{11} - \frac{2}{5} a^{9} - \frac{12}{25} a^{8} - \frac{8}{25} a^{7} - \frac{11}{25} a^{6} + \frac{3}{25} a^{5} + \frac{11}{25} a^{4} + \frac{4}{25} a^{3} - \frac{11}{25} a^{2} + \frac{1}{5}$, $\frac{1}{25025} a^{18} + \frac{356}{25025} a^{17} + \frac{307}{25025} a^{16} - \frac{201}{25025} a^{15} - \frac{401}{25025} a^{14} + \frac{157}{25025} a^{13} + \frac{354}{25025} a^{12} + \frac{1753}{25025} a^{11} - \frac{4}{3575} a^{10} - \frac{534}{1925} a^{9} + \frac{6091}{25025} a^{8} - \frac{3008}{25025} a^{7} - \frac{28}{65} a^{6} - \frac{10056}{25025} a^{5} + \frac{11488}{25025} a^{4} - \frac{288}{1925} a^{3} - \frac{7417}{25025} a^{2} - \frac{383}{5005} a - \frac{2368}{5005}$, $\frac{1}{14965277239105652333875} a^{19} - \frac{164502684068795143}{14965277239105652333875} a^{18} - \frac{58007958300581427449}{2993055447821130466775} a^{17} + \frac{2233983358846073603}{2993055447821130466775} a^{16} + \frac{16010411247568682976}{1360479749009604757625} a^{15} + \frac{824906870111117132552}{14965277239105652333875} a^{14} - \frac{8839320322073250274}{598611089564226093355} a^{13} + \frac{2643981489041339677}{598611089564226093355} a^{12} + \frac{44420742561031539873}{2137896748443664619125} a^{11} - \frac{501702679450262402103}{14965277239105652333875} a^{10} - \frac{775194903347827821387}{2993055447821130466775} a^{9} + \frac{56538445545766154129}{272095949801920951525} a^{8} + \frac{45264955829597993526}{164453596034128047625} a^{7} + \frac{509511841159825082919}{1151175172238896333375} a^{6} - \frac{1368533532600503902}{54419189960384190305} a^{5} + \frac{117928641353023330461}{598611089564226093355} a^{4} - \frac{301297311644553046124}{1360479749009604757625} a^{3} - \frac{6978799849252495324078}{14965277239105652333875} a^{2} + \frac{230100168593772826361}{2993055447821130466775} a - \frac{146771935758056816761}{427579349688732923825}$

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

Class group and class number

$C_{4}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 123933.986843 \)
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

\(\Q(\sqrt{-455}) \), \(\Q(\sqrt{-91}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-91})\), 5.1.207025.1 x5, 10.0.19501004534375.1, 10.0.3900200906875.1 x5, 10.2.214296753125.1 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.4.2.1$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}$
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}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$