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Properties

Label 20.12.1753321685...3125.3
Degree $20$
Signature $[12, 4]$
Discriminant $3^{10}\cdot 5^{11}\cdot 239^{10}$
Root discriminant $64.89$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![243241, -669901, -1168449, 2415120, 1266084, -3267388, 102034, 1757562, -607343, -255161, 164539, -9343, 34, -10945, 5138, 91, -635, 168, 2, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 2*x^18 + 168*x^17 - 635*x^16 + 91*x^15 + 5138*x^14 - 10945*x^13 + 34*x^12 - 9343*x^11 + 164539*x^10 - 255161*x^9 - 607343*x^8 + 1757562*x^7 + 102034*x^6 - 3267388*x^5 + 1266084*x^4 + 2415120*x^3 - 1168449*x^2 - 669901*x + 243241)
 
gp: K = bnfinit(x^20 - 8*x^19 + 2*x^18 + 168*x^17 - 635*x^16 + 91*x^15 + 5138*x^14 - 10945*x^13 + 34*x^12 - 9343*x^11 + 164539*x^10 - 255161*x^9 - 607343*x^8 + 1757562*x^7 + 102034*x^6 - 3267388*x^5 + 1266084*x^4 + 2415120*x^3 - 1168449*x^2 - 669901*x + 243241, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 2 x^{18} + 168 x^{17} - 635 x^{16} + 91 x^{15} + 5138 x^{14} - 10945 x^{13} + 34 x^{12} - 9343 x^{11} + 164539 x^{10} - 255161 x^{9} - 607343 x^{8} + 1757562 x^{7} + 102034 x^{6} - 3267388 x^{5} + 1266084 x^{4} + 2415120 x^{3} - 1168449 x^{2} - 669901 x + 243241 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1753321685810638349237472141064453125=3^{10}\cdot 5^{11}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.89$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{15} a^{16} - \frac{2}{15} a^{14} - \frac{1}{15} a^{13} - \frac{1}{15} a^{12} + \frac{1}{5} a^{11} + \frac{1}{15} a^{10} + \frac{2}{15} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{15} a^{5} - \frac{1}{15} a^{4} - \frac{1}{5} a^{3} - \frac{1}{15} a^{2} + \frac{1}{15} a - \frac{4}{15}$, $\frac{1}{15} a^{17} - \frac{2}{15} a^{15} - \frac{1}{15} a^{14} - \frac{1}{15} a^{13} + \frac{1}{5} a^{12} + \frac{1}{15} a^{11} + \frac{2}{15} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{15} a^{6} - \frac{1}{15} a^{5} - \frac{1}{5} a^{4} - \frac{1}{15} a^{3} + \frac{1}{15} a^{2} - \frac{4}{15} a$, $\frac{1}{75} a^{18} + \frac{1}{75} a^{17} + \frac{2}{75} a^{16} + \frac{2}{75} a^{15} - \frac{2}{15} a^{14} + \frac{8}{75} a^{13} + \frac{8}{75} a^{11} + \frac{2}{25} a^{10} + \frac{14}{75} a^{9} + \frac{12}{25} a^{8} + \frac{4}{25} a^{7} - \frac{34}{75} a^{6} - \frac{2}{75} a^{5} - \frac{1}{25} a^{4} - \frac{22}{75} a^{3} - \frac{37}{75} a^{2} - \frac{1}{5} a + \frac{8}{25}$, $\frac{1}{2643834756319324961092403119428021624279608687437689675} a^{19} - \frac{260808155260857968558219977703932212135868014399421}{881278252106441653697467706476007208093202895812563225} a^{18} - \frac{23172437634572963635399509202121076999927715178468377}{2643834756319324961092403119428021624279608687437689675} a^{17} + \frac{86654359629079400375383366268619893337792711028162729}{2643834756319324961092403119428021624279608687437689675} a^{16} - \frac{93884126335116156311597454335967347940050619438847336}{881278252106441653697467706476007208093202895812563225} a^{15} - \frac{43173454476494327546838860316398954646085611776548472}{2643834756319324961092403119428021624279608687437689675} a^{14} - \frac{966282432964161466825863593798850004963422895266524627}{2643834756319324961092403119428021624279608687437689675} a^{13} - \frac{856710956364438283055781289463335493835302640344006792}{2643834756319324961092403119428021624279608687437689675} a^{12} + \frac{19303815804557712385057388400179866615777057310431973}{881278252106441653697467706476007208093202895812563225} a^{11} - \frac{177489158093876863529341721031944660969258337987139628}{528766951263864992218480623885604324855921737487537935} a^{10} - \frac{249758419433168815455264397177743870796407933079822538}{528766951263864992218480623885604324855921737487537935} a^{9} + \frac{563693512045660319363433394735544580703128136182190388}{2643834756319324961092403119428021624279608687437689675} a^{8} + \frac{1232827734811014970293413446014842096041976009298048893}{2643834756319324961092403119428021624279608687437689675} a^{7} - \frac{105323237795168811465250242041216354268259200435140441}{2643834756319324961092403119428021624279608687437689675} a^{6} + \frac{7880811046607514987753824341991960021768451027393279}{58751883473762776913164513765067147206213526387504215} a^{5} - \frac{160936697509354185832393699488514402165767067049572778}{528766951263864992218480623885604324855921737487537935} a^{4} + \frac{1043089902695452093159598984006037418583951548259645296}{2643834756319324961092403119428021624279608687437689675} a^{3} - \frac{225698983342418031467910340775420011123104264333178642}{2643834756319324961092403119428021624279608687437689675} a^{2} + \frac{487408034820062811800817926550048404987884134456449674}{2643834756319324961092403119428021624279608687437689675} a + \frac{207479845310144022827682664112803576162255851940597744}{2643834756319324961092403119428021624279608687437689675}$

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:  $15$
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:  \( 130768352923 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed