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Properties

Label 20.0.10850417850...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{4}\cdot 5^{17}\cdot 23^{4}\cdot 89^{4}$
Root discriminant $22.48$
Ramified primes $3, 5, 23, 89$
Class number $2$
Class group $[2]$
Galois group $C_4\times S_5$ (as 20T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, 75, -75, 200, -225, 660, -1125, 1900, -1910, 1640, -1005, 641, -345, 242, -120, 79, -40, 18, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 18*x^18 - 40*x^17 + 79*x^16 - 120*x^15 + 242*x^14 - 345*x^13 + 641*x^12 - 1005*x^11 + 1640*x^10 - 1910*x^9 + 1900*x^8 - 1125*x^7 + 660*x^6 - 225*x^5 + 200*x^4 - 75*x^3 + 75*x^2 + 25)
 
gp: K = bnfinit(x^20 - 5*x^19 + 18*x^18 - 40*x^17 + 79*x^16 - 120*x^15 + 242*x^14 - 345*x^13 + 641*x^12 - 1005*x^11 + 1640*x^10 - 1910*x^9 + 1900*x^8 - 1125*x^7 + 660*x^6 - 225*x^5 + 200*x^4 - 75*x^3 + 75*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 18 x^{18} - 40 x^{17} + 79 x^{16} - 120 x^{15} + 242 x^{14} - 345 x^{13} + 641 x^{12} - 1005 x^{11} + 1640 x^{10} - 1910 x^{9} + 1900 x^{8} - 1125 x^{7} + 660 x^{6} - 225 x^{5} + 200 x^{4} - 75 x^{3} + 75 x^{2} + 25 \)

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:  \(1085041785093811798095703125=3^{4}\cdot 5^{17}\cdot 23^{4}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.48$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{12} - \frac{1}{5} a^{10} + \frac{2}{5} a^{8} + \frac{1}{5} a^{6}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{13} - \frac{1}{5} a^{11} + \frac{2}{5} a^{9} + \frac{1}{5} a^{7}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{6}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{7}$, $\frac{1}{1349495645} a^{18} - \frac{8634994}{1349495645} a^{17} - \frac{17751374}{269899129} a^{16} - \frac{13989841}{1349495645} a^{15} + \frac{62979927}{1349495645} a^{14} + \frac{217040952}{1349495645} a^{13} + \frac{344798981}{1349495645} a^{12} + \frac{24026091}{1349495645} a^{11} + \frac{580190458}{1349495645} a^{10} + \frac{667221458}{1349495645} a^{9} + \frac{257763906}{1349495645} a^{8} - \frac{181765639}{1349495645} a^{7} - \frac{75614098}{1349495645} a^{6} - \frac{98957409}{269899129} a^{5} - \frac{133696715}{269899129} a^{4} + \frac{27107665}{269899129} a^{3} - \frac{80640554}{269899129} a^{2} - \frac{57405553}{269899129} a - \frac{127331027}{269899129}$, $\frac{1}{568785903525553975} a^{19} + \frac{34014339}{113757180705110795} a^{18} - \frac{54518616264271677}{568785903525553975} a^{17} + \frac{332260086870279}{22751436141022159} a^{16} - \frac{30687451387804306}{568785903525553975} a^{15} - \frac{6441503215256229}{113757180705110795} a^{14} - \frac{13156399014954393}{568785903525553975} a^{13} - \frac{49600288789116214}{113757180705110795} a^{12} + \frac{237096325539864561}{568785903525553975} a^{11} + \frac{42315314458899799}{113757180705110795} a^{10} + \frac{4729625644343085}{22751436141022159} a^{9} + \frac{53649572962313898}{113757180705110795} a^{8} + \frac{14442448410131803}{113757180705110795} a^{7} + \frac{42822147639436151}{113757180705110795} a^{6} + \frac{41754312298207501}{113757180705110795} a^{5} + \frac{9381325166739161}{22751436141022159} a^{4} - \frac{1535491172829601}{22751436141022159} a^{3} + \frac{750611618940513}{22751436141022159} a^{2} + \frac{678937280753706}{22751436141022159} a + \frac{208026317488520}{22751436141022159}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{526709356442193}{113757180705110795} a^{19} - \frac{480639196888279}{22751436141022159} a^{18} + \frac{8319278728144466}{113757180705110795} a^{17} - \frac{16905463858572337}{113757180705110795} a^{16} + \frac{32078019265936643}{113757180705110795} a^{15} - \frac{44095553123255988}{113757180705110795} a^{14} + \frac{96946848293554989}{113757180705110795} a^{13} - \frac{122425219848288339}{113757180705110795} a^{12} + \frac{251258954426526997}{113757180705110795} a^{11} - \frac{379168228525962452}{113757180705110795} a^{10} + \frac{607098133516886477}{113757180705110795} a^{9} - \frac{622640460205027811}{113757180705110795} a^{8} + \frac{101701611501917285}{22751436141022159} a^{7} - \frac{111285448525487337}{113757180705110795} a^{6} + \frac{1008631017193228}{22751436141022159} a^{5} - \frac{3428349719463868}{22751436141022159} a^{4} + \frac{29773512852415748}{22751436141022159} a^{3} - \frac{56202901380168562}{22751436141022159} a^{2} + \frac{12068114082782085}{22751436141022159} a + \frac{1698424203409379}{22751436141022159} \) (order $10$)
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:  \( 366014.001413 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times S_5$ (as 20T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 28 conjugacy class representatives for $C_4\times S_5$
Character table for $C_4\times S_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.767625.1, 10.10.2946240703125.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 $20$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$