↑

Properties

Label 20.20.4132640411...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{26}\cdot 11^{4}\cdot 29^{5}\cdot 31^{4}$
Root discriminant $60.37$
Ramified primes $5, 11, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T299

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![139, 475, -13110, -35290, 142945, 285219, -338365, -639230, 306525, 558275, -163635, -233775, 56205, 49775, -11255, -5278, 1145, 265, -55, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 55*x^18 + 265*x^17 + 1145*x^16 - 5278*x^15 - 11255*x^14 + 49775*x^13 + 56205*x^12 - 233775*x^11 - 163635*x^10 + 558275*x^9 + 306525*x^8 - 639230*x^7 - 338365*x^6 + 285219*x^5 + 142945*x^4 - 35290*x^3 - 13110*x^2 + 475*x + 139)
 
gp: K = bnfinit(x^20 - 5*x^19 - 55*x^18 + 265*x^17 + 1145*x^16 - 5278*x^15 - 11255*x^14 + 49775*x^13 + 56205*x^12 - 233775*x^11 - 163635*x^10 + 558275*x^9 + 306525*x^8 - 639230*x^7 - 338365*x^6 + 285219*x^5 + 142945*x^4 - 35290*x^3 - 13110*x^2 + 475*x + 139, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 55 x^{18} + 265 x^{17} + 1145 x^{16} - 5278 x^{15} - 11255 x^{14} + 49775 x^{13} + 56205 x^{12} - 233775 x^{11} - 163635 x^{10} + 558275 x^{9} + 306525 x^{8} - 639230 x^{7} - 338365 x^{6} + 285219 x^{5} + 142945 x^{4} - 35290 x^{3} - 13110 x^{2} + 475 x + 139 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(413264041171139760315418243408203125=5^{26}\cdot 11^{4}\cdot 29^{5}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.37$
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:  $5, 11, 29, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{33} a^{18} - \frac{4}{33} a^{17} + \frac{4}{33} a^{16} - \frac{16}{33} a^{15} - \frac{16}{33} a^{14} + \frac{1}{33} a^{13} - \frac{8}{33} a^{12} - \frac{1}{3} a^{11} - \frac{1}{11} a^{10} + \frac{16}{33} a^{9} - \frac{7}{33} a^{8} - \frac{8}{33} a^{7} - \frac{10}{33} a^{6} + \frac{5}{33} a^{5} - \frac{1}{11} a^{4} - \frac{7}{33} a^{3} - \frac{3}{11} a^{2} - \frac{4}{11} a - \frac{5}{33}$, $\frac{1}{482606207696565385047693134695966379782428699} a^{19} + \frac{4982387895362027565082222543741382232347449}{482606207696565385047693134695966379782428699} a^{18} + \frac{657426748097137511237088492626618301396772}{11770883114550375245065686212096740970303139} a^{17} - \frac{4908137001892192180628538947393649741113647}{160868735898855128349231044898655459927476233} a^{16} - \frac{71731037120345546183539187243065750683940922}{160868735898855128349231044898655459927476233} a^{15} + \frac{19917036323548527237364133764931497024826181}{160868735898855128349231044898655459927476233} a^{14} - \frac{59542517907459261034032184307564790629089408}{482606207696565385047693134695966379782428699} a^{13} - \frac{79081731407665077864662370162498261444350138}{482606207696565385047693134695966379782428699} a^{12} - \frac{198527262092301214050670943384722117737774338}{482606207696565385047693134695966379782428699} a^{11} + \frac{31385899828546997259161376345759510364447498}{160868735898855128349231044898655459927476233} a^{10} - \frac{145117691158805851767934518556471426008996560}{482606207696565385047693134695966379782428699} a^{9} + \frac{203350086210408825786340550723988572328214381}{482606207696565385047693134695966379782428699} a^{8} + \frac{151850550113729299072063076300871558308788241}{482606207696565385047693134695966379782428699} a^{7} - \frac{15978093997957391326288814395938732113212085}{68943743956652197863956162099423768540346957} a^{6} + \frac{6449741492822975764763578289432426218418449}{43873291608778671367972103154178761798402609} a^{5} - \frac{2534689762837027317371858288197359478956260}{6267613086968381623996014736311251685486087} a^{4} - \frac{1283760372828522813321129079114373906596604}{14624430536259557122657367718059587266134203} a^{3} - \frac{232409420618273242796878513262155096274049355}{482606207696565385047693134695966379782428699} a^{2} + \frac{615357052743310662654739481483815651652221}{12374518146065779103787003453742727686728941} a + \frac{46017144906848834503423847808346960786755830}{160868735898855128349231044898655459927476233}$

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:  $19$
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:  \( 304766298548 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T299:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5000
The 230 conjugacy class representatives for t20n299 are not computed
Character table for t20n299 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{10}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.4.2$x^{5} - 891$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$29$29.10.5.2$x^{10} - 707281 x^{2} + 225622639$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.5.4.5$x^{5} - 74431$$5$$1$$4$$C_5$$[\ ]_{5}$
31.10.0.1$x^{10} - x + 11$$1$$10$$0$$C_{10}$$[\ ]^{10}$