↑

Properties

Label 20.0.24217697846...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{13}\cdot 269^{10}$
Root discriminant $46.69$
Ramified primes $5, 269$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:S_5$ (as 20T120)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10555, 9565, 5468, 66361, 93564, 45116, 71096, 28625, 14031, 9176, 6012, -4335, 5536, -3550, 1894, -841, 286, -86, 23, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 23*x^18 - 86*x^17 + 286*x^16 - 841*x^15 + 1894*x^14 - 3550*x^13 + 5536*x^12 - 4335*x^11 + 6012*x^10 + 9176*x^9 + 14031*x^8 + 28625*x^7 + 71096*x^6 + 45116*x^5 + 93564*x^4 + 66361*x^3 + 5468*x^2 + 9565*x + 10555)
 
gp: K = bnfinit(x^20 - 4*x^19 + 23*x^18 - 86*x^17 + 286*x^16 - 841*x^15 + 1894*x^14 - 3550*x^13 + 5536*x^12 - 4335*x^11 + 6012*x^10 + 9176*x^9 + 14031*x^8 + 28625*x^7 + 71096*x^6 + 45116*x^5 + 93564*x^4 + 66361*x^3 + 5468*x^2 + 9565*x + 10555, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 23 x^{18} - 86 x^{17} + 286 x^{16} - 841 x^{15} + 1894 x^{14} - 3550 x^{13} + 5536 x^{12} - 4335 x^{11} + 6012 x^{10} + 9176 x^{9} + 14031 x^{8} + 28625 x^{7} + 71096 x^{6} + 45116 x^{5} + 93564 x^{4} + 66361 x^{3} + 5468 x^{2} + 9565 x + 10555 \)

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:  \(2421769784649371094992432861328125=5^{13}\cdot 269^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.69$
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:  $5, 269$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{15} - \frac{1}{5} a^{13} + \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{16} - \frac{1}{5} a^{14} + \frac{2}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{2687648728310163765350177756610247989409047922925} a^{19} - \frac{136446966813494722383671227297185413056462480866}{2687648728310163765350177756610247989409047922925} a^{18} + \frac{23489968815945772534535096338693538869031778189}{537529745662032753070035551322049597881809584585} a^{17} + \frac{562863425247143057460649532364248059759338745149}{2687648728310163765350177756610247989409047922925} a^{16} + \frac{420493041558183845879549594408394246778417597103}{2687648728310163765350177756610247989409047922925} a^{15} + \frac{989996026218356635032032742189583235139396320598}{2687648728310163765350177756610247989409047922925} a^{14} - \frac{1106133287094533485478608105386432132021562782062}{2687648728310163765350177756610247989409047922925} a^{13} - \frac{1325671827624762351782073570280732573429372785596}{2687648728310163765350177756610247989409047922925} a^{12} + \frac{1112227743205982289532696743079534806006324350933}{2687648728310163765350177756610247989409047922925} a^{11} + \frac{1243470746611956346215129550044064660763052206044}{2687648728310163765350177756610247989409047922925} a^{10} + \frac{736304165750123263058627844060019696519295329594}{2687648728310163765350177756610247989409047922925} a^{9} - \frac{618717073730731627135369450865350511236605083972}{2687648728310163765350177756610247989409047922925} a^{8} + \frac{209205266052641991242064397259244196812093701891}{537529745662032753070035551322049597881809584585} a^{7} + \frac{3728027912237041276686756793183001614952725504}{537529745662032753070035551322049597881809584585} a^{6} + \frac{1180199084202430570650654530824715579639462876521}{2687648728310163765350177756610247989409047922925} a^{5} - \frac{305898527009988787865825939070306791284946047716}{2687648728310163765350177756610247989409047922925} a^{4} - \frac{733535881768292396030265792849169395132933089754}{2687648728310163765350177756610247989409047922925} a^{3} - \frac{102012386516206484657228925146146257184752231211}{2687648728310163765350177756610247989409047922925} a^{2} + \frac{37019118326887723340418850284188889804104636559}{107505949132406550614007110264409919576361916917} a - \frac{219958969270494772869064075029992536714567744877}{537529745662032753070035551322049597881809584585}$

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:  \( 472140760.692 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:S_5$ (as 20T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 19 conjugacy class representatives for $C_4:S_5$
Character table for $C_4:S_5$

Intermediate fields

\(\Q(\sqrt{269}) \), 4.0.361805.1, 5.1.33625.1, 10.2.22008043005453125.1

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 $20$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
269Data not computed