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Properties

Label 20.0.53430555565...8864.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{19}\cdot 547^{9}$
Root discriminant $96.91$
Ramified primes $2, 3, 547$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_5:S_4$ (as 20T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10205433, 5302152, 11292687, 2519982, 14258529, 8671992, 1511325, -2650572, 886820, -171860, -19094, -58040, 40588, -116, -2520, -772, 479, 26, -19, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 19*x^18 + 26*x^17 + 479*x^16 - 772*x^15 - 2520*x^14 - 116*x^13 + 40588*x^12 - 58040*x^11 - 19094*x^10 - 171860*x^9 + 886820*x^8 - 2650572*x^7 + 1511325*x^6 + 8671992*x^5 + 14258529*x^4 + 2519982*x^3 + 11292687*x^2 + 5302152*x + 10205433)
 
gp: K = bnfinit(x^20 - 4*x^19 - 19*x^18 + 26*x^17 + 479*x^16 - 772*x^15 - 2520*x^14 - 116*x^13 + 40588*x^12 - 58040*x^11 - 19094*x^10 - 171860*x^9 + 886820*x^8 - 2650572*x^7 + 1511325*x^6 + 8671992*x^5 + 14258529*x^4 + 2519982*x^3 + 11292687*x^2 + 5302152*x + 10205433, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 19 x^{18} + 26 x^{17} + 479 x^{16} - 772 x^{15} - 2520 x^{14} - 116 x^{13} + 40588 x^{12} - 58040 x^{11} - 19094 x^{10} - 171860 x^{9} + 886820 x^{8} - 2650572 x^{7} + 1511325 x^{6} + 8671992 x^{5} + 14258529 x^{4} + 2519982 x^{3} + 11292687 x^{2} + 5302152 x + 10205433 \)

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:  \(5343055556521822422488603677866788388864=2^{20}\cdot 3^{19}\cdot 547^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.91$
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:  $2, 3, 547$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{3} a^{8} + \frac{2}{9} a^{7} - \frac{2}{9} a^{6} + \frac{4}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{1}{9} a^{10} - \frac{4}{9} a^{8} - \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{2}{9} a^{5} + \frac{4}{9} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{4923} a^{16} - \frac{22}{547} a^{15} - \frac{185}{4923} a^{14} + \frac{188}{4923} a^{13} - \frac{562}{4923} a^{12} - \frac{749}{4923} a^{11} - \frac{722}{4923} a^{10} + \frac{53}{1641} a^{9} + \frac{695}{4923} a^{8} - \frac{728}{4923} a^{7} - \frac{2360}{4923} a^{6} + \frac{1127}{4923} a^{5} + \frac{649}{4923} a^{4} + \frac{362}{1641} a^{3} - \frac{114}{547} a^{2} - \frac{667}{1641} a - \frac{235}{547}$, $\frac{1}{103383} a^{17} + \frac{1}{11487} a^{16} + \frac{4777}{103383} a^{15} + \frac{1702}{34461} a^{14} - \frac{4859}{103383} a^{13} + \frac{650}{4923} a^{12} + \frac{7241}{103383} a^{11} - \frac{4364}{34461} a^{10} - \frac{5776}{103383} a^{9} - \frac{432}{3829} a^{8} + \frac{6668}{103383} a^{7} - \frac{3652}{34461} a^{6} - \frac{12458}{34461} a^{5} + \frac{461}{11487} a^{4} + \frac{83}{3829} a^{3} + \frac{2071}{11487} a^{2} + \frac{2425}{11487} a + \frac{106}{547}$, $\frac{1}{103383} a^{18} - \frac{8}{103383} a^{16} - \frac{827}{34461} a^{15} + \frac{550}{14769} a^{14} + \frac{31}{34461} a^{13} - \frac{10588}{103383} a^{12} - \frac{355}{34461} a^{11} + \frac{4805}{103383} a^{10} + \frac{85}{547} a^{9} + \frac{1289}{103383} a^{8} + \frac{5032}{11487} a^{7} - \frac{16214}{34461} a^{6} - \frac{1570}{4923} a^{5} + \frac{223}{11487} a^{4} - \frac{3}{7} a^{3} - \frac{4531}{11487} a^{2} - \frac{1248}{3829} a + \frac{268}{547}$, $\frac{1}{65639637187475480814792710510639634910456283452184298338271} a^{19} - \frac{48133002107775688490424094297787882787126421873486334}{21879879062491826938264236836879878303485427817394766112757} a^{18} + \frac{48346188125899236116540280891371611538975876830700635}{65639637187475480814792710510639634910456283452184298338271} a^{17} + \frac{487226246117275966094703789196562111747521957026849329}{21879879062491826938264236836879878303485427817394766112757} a^{16} + \frac{2414900002310336002819061099212858149827147869747033954786}{65639637187475480814792710510639634910456283452184298338271} a^{15} + \frac{24227183611737953613811889283412795984974116795024155079}{2431097673610202993140470759653319811498380868599418456973} a^{14} - \frac{230281457428664344008430252978511932166914637576542848247}{9377091026782211544970387215805662130065183350312042619753} a^{13} + \frac{1121878376250389375613099242443411335525034179159185838323}{7293293020830608979421412278959959434495142605798255370919} a^{12} - \frac{9785841329347567764906339301218054448833710801762416317103}{65639637187475480814792710510639634910456283452184298338271} a^{11} - \frac{232630917334527724968769370868773238547354846004346147889}{3125697008927403848323462405268554043355061116770680873251} a^{10} - \frac{1572599045114311676147652915470824166627430665723220307558}{65639637187475480814792710510639634910456283452184298338271} a^{9} + \frac{7081036584199336544296593034497856631650796428157540112055}{21879879062491826938264236836879878303485427817394766112757} a^{8} - \frac{1114868532052983661496572888058464122495728961434851669731}{2431097673610202993140470759653319811498380868599418456973} a^{7} + \frac{1173654203643586516051786364114324495769146461886584176613}{7293293020830608979421412278959959434495142605798255370919} a^{6} - \frac{191498162001255728586360280819904242138775668294191623521}{3125697008927403848323462405268554043355061116770680873251} a^{5} - \frac{811795496906781529170261427278833131040252351401165342624}{2431097673610202993140470759653319811498380868599418456973} a^{4} + \frac{991860234738188825220572243990319084345821418854030749547}{2431097673610202993140470759653319811498380868599418456973} a^{3} - \frac{3122475238586044650295112214564887849621892783182791375370}{7293293020830608979421412278959959434495142605798255370919} a^{2} + \frac{642901734157662646421397210091611240590713304361757215872}{2431097673610202993140470759653319811498380868599418456973} a - \frac{85678993741546249683578464947488564563473436560602053275}{347299667658600427591495822807617115928340124085631208139}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 2335295541180 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:S_4$ (as 20T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 13 conjugacy class representatives for $C_5:S_4$
Character table for $C_5:S_4$

Intermediate fields

4.0.236304.1, 5.5.2692881.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 sibling: 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 R R $15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ $15{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
2Data not computed
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
547Data not computed