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Properties

Label 20.0.61389740579...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 239^{8}$
Root discriminant $34.63$
Ramified primes $3, 5, 239$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_2^2\times D_5$ (as 20T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41491, -137631, 225474, -231346, 177709, -83653, 6725, 37170, -37870, 26305, -9342, 1477, 2256, -1936, 1335, -593, 263, -81, 26, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 26*x^18 - 81*x^17 + 263*x^16 - 593*x^15 + 1335*x^14 - 1936*x^13 + 2256*x^12 + 1477*x^11 - 9342*x^10 + 26305*x^9 - 37870*x^8 + 37170*x^7 + 6725*x^6 - 83653*x^5 + 177709*x^4 - 231346*x^3 + 225474*x^2 - 137631*x + 41491)
 
gp: K = bnfinit(x^20 - 5*x^19 + 26*x^18 - 81*x^17 + 263*x^16 - 593*x^15 + 1335*x^14 - 1936*x^13 + 2256*x^12 + 1477*x^11 - 9342*x^10 + 26305*x^9 - 37870*x^8 + 37170*x^7 + 6725*x^6 - 83653*x^5 + 177709*x^4 - 231346*x^3 + 225474*x^2 - 137631*x + 41491, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 26 x^{18} - 81 x^{17} + 263 x^{16} - 593 x^{15} + 1335 x^{14} - 1936 x^{13} + 2256 x^{12} + 1477 x^{11} - 9342 x^{10} + 26305 x^{9} - 37870 x^{8} + 37170 x^{7} + 6725 x^{6} - 83653 x^{5} + 177709 x^{4} - 231346 x^{3} + 225474 x^{2} - 137631 x + 41491 \)

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:  \(6138974057914386475157900390625=3^{10}\cdot 5^{10}\cdot 239^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.63$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{15} a^{16} - \frac{2}{15} a^{15} - \frac{2}{15} a^{14} + \frac{2}{5} a^{13} - \frac{7}{15} a^{12} - \frac{1}{15} a^{11} + \frac{1}{5} a^{10} - \frac{2}{15} a^{9} - \frac{7}{15} a^{8} + \frac{1}{15} a^{7} - \frac{7}{15} a^{5} - \frac{4}{15} a^{4} + \frac{1}{5} a^{3} + \frac{4}{15} a^{2} - \frac{1}{3} a + \frac{4}{15}$, $\frac{1}{15} a^{17} - \frac{1}{15} a^{15} + \frac{2}{15} a^{14} - \frac{1}{3} a^{12} + \frac{1}{15} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{15} a^{8} + \frac{2}{15} a^{7} + \frac{1}{5} a^{6} + \frac{7}{15} a^{5} - \frac{1}{3} a^{4} - \frac{7}{15} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{8354107575} a^{18} + \frac{23710939}{928234175} a^{17} - \frac{16530958}{928234175} a^{16} - \frac{154019123}{928234175} a^{15} - \frac{33741131}{759464325} a^{14} - \frac{223905011}{759464325} a^{13} - \frac{33534694}{2784702525} a^{12} - \frac{2976961619}{8354107575} a^{11} - \frac{540301691}{1670821515} a^{10} - \frac{930180164}{2784702525} a^{9} + \frac{1365405416}{8354107575} a^{8} - \frac{2426448016}{8354107575} a^{7} - \frac{595325147}{1670821515} a^{6} + \frac{35951406}{84384925} a^{5} - \frac{3480624116}{8354107575} a^{4} - \frac{62228296}{1670821515} a^{3} - \frac{2150580107}{8354107575} a^{2} + \frac{597624107}{8354107575} a + \frac{2412501224}{8354107575}$, $\frac{1}{5086341342972686994720707914950825} a^{19} + \frac{251723493059020476759983}{5086341342972686994720707914950825} a^{18} - \frac{2174353525403695215600535224704}{67817884572969159929609438866011} a^{17} - \frac{452179112590947039103038520931}{19487897865795735611956735306325} a^{16} - \frac{160874765958908635829181813662339}{1017268268594537398944141582990165} a^{15} - \frac{7190254652962933687100325837582}{51377185282552393886067756716675} a^{14} - \frac{923125099127955743660204681591459}{5086341342972686994720707914950825} a^{13} - \frac{14283004645273606568907739343677}{175391080792161620507610617756925} a^{12} - \frac{487843296500117747429914654375793}{5086341342972686994720707914950825} a^{11} - \frac{102258492131934584143648348820737}{462394667542971544974609810450075} a^{10} - \frac{2413191830559035820046604772683083}{5086341342972686994720707914950825} a^{9} - \frac{129402504172883133040623556237246}{565149038108076332746745323883425} a^{8} + \frac{744106285590999484676864143622476}{1695447114324228998240235971650275} a^{7} - \frac{1110015607854456717148166667458306}{5086341342972686994720707914950825} a^{6} + \frac{902735415687503034469047253813432}{5086341342972686994720707914950825} a^{5} - \frac{170879717136737233751882810651774}{1695447114324228998240235971650275} a^{4} + \frac{842336732216806001085504328544456}{1695447114324228998240235971650275} a^{3} - \frac{98888444399485982659714170224792}{5086341342972686994720707914950825} a^{2} - \frac{647684745482070564233864433816089}{1695447114324228998240235971650275} a + \frac{112114180228814393358899268584443}{5086341342972686994720707914950825}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( -\frac{42888313599281}{443715330654973575} a^{19} + \frac{59326944028577}{147905110218324525} a^{18} - \frac{320275703829502}{147905110218324525} a^{17} + \frac{293698848085067}{49301703406108175} a^{16} - \frac{8937566900405788}{443715330654973575} a^{15} + \frac{1599093722097094}{40337757332270325} a^{14} - \frac{918404524410863}{9860340681221635} a^{13} + \frac{9221677622460551}{88743066130994715} a^{12} - \frac{52312975825213423}{443715330654973575} a^{11} - \frac{37522416742729861}{147905110218324525} a^{10} + \frac{12604668517480697}{17748613226198943} a^{9} - \frac{845070572371436812}{443715330654973575} a^{8} + \frac{876416543745202118}{443715330654973575} a^{7} - \frac{238320418774658003}{147905110218324525} a^{6} - \frac{1006222003298799251}{443715330654973575} a^{5} + \frac{2934499512259845688}{443715330654973575} a^{4} - \frac{4989558997514662268}{443715330654973575} a^{3} + \frac{5360638210882002674}{443715330654973575} a^{2} - \frac{838349183816645342}{88743066130994715} a + \frac{544774709598063346}{147905110218324525} \) (order $6$)
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:  \( 10123076.9997 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_5$ (as 20T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_2^2\times D_5$
Character table for $C_2^2\times D_5$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 5.5.12852225.1, 10.0.2477695311759375.1, 10.10.825898437253125.1, 10.0.495539062351875.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
239Data not computed