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Properties

Label 20.0.27460473324...5625.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{20}\cdot 5^{10}\cdot 73^{8}$
Root discriminant $37.32$
Ramified primes $3, 5, 73$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, 759, -4030, -6303, 38958, 1167, 21891, 1950, 9858, -6060, 1077, 744, -267, 951, -132, 60, -48, -12, 14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 14*x^18 - 12*x^17 - 48*x^16 + 60*x^15 - 132*x^14 + 951*x^13 - 267*x^12 + 744*x^11 + 1077*x^10 - 6060*x^9 + 9858*x^8 + 1950*x^7 + 21891*x^6 + 1167*x^5 + 38958*x^4 - 6303*x^3 - 4030*x^2 + 759*x + 529)
 
gp: K = bnfinit(x^20 - 6*x^19 + 14*x^18 - 12*x^17 - 48*x^16 + 60*x^15 - 132*x^14 + 951*x^13 - 267*x^12 + 744*x^11 + 1077*x^10 - 6060*x^9 + 9858*x^8 + 1950*x^7 + 21891*x^6 + 1167*x^5 + 38958*x^4 - 6303*x^3 - 4030*x^2 + 759*x + 529, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 14 x^{18} - 12 x^{17} - 48 x^{16} + 60 x^{15} - 132 x^{14} + 951 x^{13} - 267 x^{12} + 744 x^{11} + 1077 x^{10} - 6060 x^{9} + 9858 x^{8} + 1950 x^{7} + 21891 x^{6} + 1167 x^{5} + 38958 x^{4} - 6303 x^{3} - 4030 x^{2} + 759 x + 529 \)

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:  \(27460473324661212646782041015625=3^{20}\cdot 5^{10}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.32$
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:  $3, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{14} + \frac{1}{9} a^{12} - \frac{2}{9} a^{10} + \frac{1}{9} a^{8} + \frac{4}{9} a^{6} + \frac{4}{9} a^{4} + \frac{1}{9} a^{2} - \frac{2}{9}$, $\frac{1}{621} a^{17} - \frac{10}{621} a^{16} - \frac{5}{621} a^{15} + \frac{1}{27} a^{14} - \frac{29}{621} a^{13} + \frac{38}{621} a^{12} + \frac{70}{621} a^{11} - \frac{7}{621} a^{10} - \frac{275}{621} a^{9} - \frac{112}{621} a^{8} + \frac{109}{621} a^{7} + \frac{89}{621} a^{6} - \frac{149}{621} a^{5} + \frac{239}{621} a^{4} - \frac{266}{621} a^{3} - \frac{94}{621} a^{2} + \frac{58}{621} a + \frac{1}{27}$, $\frac{1}{9789462009} a^{18} - \frac{4086944}{9789462009} a^{17} - \frac{400082893}{9789462009} a^{16} + \frac{96899734}{9789462009} a^{15} + \frac{493883939}{9789462009} a^{14} - \frac{1479441320}{9789462009} a^{13} + \frac{785530070}{9789462009} a^{12} - \frac{3562510367}{9789462009} a^{11} - \frac{3828317995}{9789462009} a^{10} + \frac{2529207451}{9789462009} a^{9} + \frac{2926856453}{9789462009} a^{8} - \frac{428536976}{9789462009} a^{7} - \frac{22136491}{9789462009} a^{6} + \frac{99205586}{425628783} a^{5} - \frac{2509903948}{9789462009} a^{4} + \frac{1160974360}{9789462009} a^{3} - \frac{834105808}{9789462009} a^{2} - \frac{4740673529}{9789462009} a - \frac{64786261}{425628783}$, $\frac{1}{2394323691246303277106570085490941} a^{19} - \frac{102947304834661064198599}{2394323691246303277106570085490941} a^{18} - \frac{94145452432113112457400612169}{798107897082101092368856695163647} a^{17} + \frac{5410449908000217113804598983120}{266035965694033697456285565054549} a^{16} - \frac{17420936895046597070459489238850}{798107897082101092368856695163647} a^{15} + \frac{8187793952771831129989357155436}{88678655231344565818761855018183} a^{14} - \frac{34712836733336221346945463261242}{798107897082101092368856695163647} a^{13} - \frac{721083235240912777293475145515}{34700343351395699668211160659289} a^{12} - \frac{96758711119714705259545731988364}{266035965694033697456285565054549} a^{11} + \frac{343104575967720356642919536321225}{798107897082101092368856695163647} a^{10} + \frac{23567884685274924596379847466627}{266035965694033697456285565054549} a^{9} + \frac{107646008621063887044394291035548}{798107897082101092368856695163647} a^{8} + \frac{215865001079187712356244882033235}{798107897082101092368856695163647} a^{7} - \frac{129160967424454948328326349502485}{266035965694033697456285565054549} a^{6} + \frac{206474424675660512337913684470373}{798107897082101092368856695163647} a^{5} + \frac{324929225697098738530552905489043}{798107897082101092368856695163647} a^{4} - \frac{107793318126649205998983544899737}{798107897082101092368856695163647} a^{3} + \frac{275902567327959412503558212504791}{798107897082101092368856695163647} a^{2} - \frac{1196457924381125390107822492490017}{2394323691246303277106570085490941} a + \frac{50225118282808822510004411157101}{104101030054187099004633481977867}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( \frac{19124807905200074475961}{151885263040705093672852929} a^{19} - \frac{145345892551045829802763}{151885263040705093672852929} a^{18} + \frac{149791700093479414666231}{50628421013568364557617643} a^{17} - \frac{216767392023838043454842}{50628421013568364557617643} a^{16} - \frac{184398864451116121681031}{50628421013568364557617643} a^{15} + \frac{862772640090463577990728}{50628421013568364557617643} a^{14} - \frac{473323049436260816135600}{16876140337856121519205881} a^{13} + \frac{7473059351011799103969086}{50628421013568364557617643} a^{12} - \frac{11441655208628541105830101}{50628421013568364557617643} a^{11} + \frac{2323133908640709830859703}{16876140337856121519205881} a^{10} - \frac{1822794416548173487773514}{50628421013568364557617643} a^{9} - \frac{16645709043511061796223121}{16876140337856121519205881} a^{8} + \frac{125127206448854849793448057}{50628421013568364557617643} a^{7} - \frac{87005364466346565062186135}{50628421013568364557617643} a^{6} + \frac{40918090803283873046464501}{16876140337856121519205881} a^{5} - \frac{230157078817131298476711841}{50628421013568364557617643} a^{4} + \frac{71337903895072008925982366}{16876140337856121519205881} a^{3} - \frac{445781342811792935808108868}{50628421013568364557617643} a^{2} + \frac{63361603728851183509424693}{151885263040705093672852929} a + \frac{6381023902766987114349266}{6603707088726308420558823} \) (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:  \( 29938408.0708 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.10791225.1, 10.2.5240274165028125.1, 10.0.349351611001875.1, 10.8.1746758055009375.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$5$5.4.2.1$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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$