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Properties

Label 20.20.4884274415...6704.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 17^{15}\cdot 53^{13}$
Root discriminant $192.50$
Ramified primes $2, 17, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:F_5$ (as 20T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-58612856, -315124556, 1644305872, -1179787402, -1813914636, 1707884929, 787082462, -826609978, -163802315, 193331394, 14939336, -24570618, -126850, 1744748, -71774, -67464, 4736, 1313, -116, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 - 116*x^18 + 1313*x^17 + 4736*x^16 - 67464*x^15 - 71774*x^14 + 1744748*x^13 - 126850*x^12 - 24570618*x^11 + 14939336*x^10 + 193331394*x^9 - 163802315*x^8 - 826609978*x^7 + 787082462*x^6 + 1707884929*x^5 - 1813914636*x^4 - 1179787402*x^3 + 1644305872*x^2 - 315124556*x - 58612856)
 
gp: K = bnfinit(x^20 - 10*x^19 - 116*x^18 + 1313*x^17 + 4736*x^16 - 67464*x^15 - 71774*x^14 + 1744748*x^13 - 126850*x^12 - 24570618*x^11 + 14939336*x^10 + 193331394*x^9 - 163802315*x^8 - 826609978*x^7 + 787082462*x^6 + 1707884929*x^5 - 1813914636*x^4 - 1179787402*x^3 + 1644305872*x^2 - 315124556*x - 58612856, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} - 116 x^{18} + 1313 x^{17} + 4736 x^{16} - 67464 x^{15} - 71774 x^{14} + 1744748 x^{13} - 126850 x^{12} - 24570618 x^{11} + 14939336 x^{10} + 193331394 x^{9} - 163802315 x^{8} - 826609978 x^{7} + 787082462 x^{6} + 1707884929 x^{5} - 1813914636 x^{4} - 1179787402 x^{3} + 1644305872 x^{2} - 315124556 x - 58612856 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4884274415246183319344904689389832005326536704=2^{16}\cdot 17^{15}\cdot 53^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $192.50$
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:  $2, 17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{3}{10} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{10} a^{11} + \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{2}{5} a^{7} + \frac{3}{10} a^{5} - \frac{2}{5} a^{3} - \frac{1}{10} a^{2} - \frac{1}{5} a$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{1}{10} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{1}{2} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{3}{10} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{9} + \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{3}{10} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{530} a^{15} - \frac{11}{530} a^{14} - \frac{21}{530} a^{13} - \frac{17}{530} a^{12} - \frac{23}{530} a^{11} + \frac{13}{530} a^{10} + \frac{61}{530} a^{9} - \frac{11}{53} a^{8} + \frac{249}{530} a^{7} - \frac{13}{106} a^{6} + \frac{59}{530} a^{5} - \frac{177}{530} a^{4} - \frac{77}{265} a^{3} - \frac{109}{530} a^{2} - \frac{3}{265} a + \frac{56}{265}$, $\frac{1}{1060} a^{16} - \frac{9}{265} a^{14} - \frac{9}{265} a^{13} + \frac{1}{530} a^{12} - \frac{7}{265} a^{11} - \frac{2}{265} a^{10} - \frac{117}{530} a^{9} + \frac{23}{530} a^{8} - \frac{20}{53} a^{7} - \frac{63}{530} a^{6} + \frac{13}{53} a^{5} - \frac{193}{1060} a^{4} - \frac{239}{530} a^{3} - \frac{41}{106} a^{2} + \frac{91}{265} a + \frac{96}{265}$, $\frac{1}{1060} a^{17} - \frac{2}{265} a^{14} - \frac{3}{265} a^{13} - \frac{1}{265} a^{12} + \frac{3}{265} a^{11} + \frac{11}{530} a^{10} + \frac{61}{530} a^{9} + \frac{23}{265} a^{8} + \frac{63}{265} a^{7} - \frac{43}{265} a^{6} + \frac{23}{1060} a^{5} + \frac{179}{530} a^{4} - \frac{9}{530} a^{3} + \frac{64}{265} a^{2} - \frac{117}{265} a - \frac{52}{265}$, $\frac{1}{1060133751779675116235140} a^{18} - \frac{89929681514794984027}{1060133751779675116235140} a^{17} - \frac{11747992861508824053}{106013375177967511623514} a^{16} + \frac{32342809742659817679}{265033437944918779058785} a^{15} + \frac{7130271090795222808212}{265033437944918779058785} a^{14} + \frac{6186710049760483105927}{530066875889837558117570} a^{13} + \frac{752013532611554231187}{53006687588983755811757} a^{12} + \frac{24546228595382657467957}{530066875889837558117570} a^{11} + \frac{18180944526611990571441}{530066875889837558117570} a^{10} + \frac{13195836117188364951047}{265033437944918779058785} a^{9} + \frac{17879480988021084895669}{106013375177967511623514} a^{8} + \frac{38471492211771182730243}{530066875889837558117570} a^{7} + \frac{46114432733805532340459}{1060133751779675116235140} a^{6} - \frac{5486061073113276902419}{20002523618484436155380} a^{5} - \frac{774736601440685545409}{53006687588983755811757} a^{4} - \frac{59404701620769219395708}{265033437944918779058785} a^{3} - \frac{41798845293631459280601}{530066875889837558117570} a^{2} - \frac{42579401412284723671021}{265033437944918779058785} a + \frac{84770423482640978786587}{265033437944918779058785}$, $\frac{1}{929512985848733506092770039860799054991020} a^{19} + \frac{6553500158736703}{464756492924366753046385019930399527495510} a^{18} + \frac{37657713279526059562886811697268823687}{464756492924366753046385019930399527495510} a^{17} - \frac{20590310996125794856965979395592188594}{232378246462183376523192509965199763747755} a^{16} + \frac{404680111638673234602905073339745038709}{464756492924366753046385019930399527495510} a^{15} + \frac{4145410980107200648183532842480740354276}{232378246462183376523192509965199763747755} a^{14} - \frac{1634199926276951899301245180209930303744}{46475649292436675304638501993039952749551} a^{13} + \frac{5238269810455241821581614375679666597491}{232378246462183376523192509965199763747755} a^{12} - \frac{3196975868049255016803395423960634850556}{232378246462183376523192509965199763747755} a^{11} - \frac{10824011755662990639947463032892399180359}{464756492924366753046385019930399527495510} a^{10} - \frac{7521687734313868447016624364822723490999}{92951298584873350609277003986079905499102} a^{9} + \frac{55709484724202073515572713998898234784237}{232378246462183376523192509965199763747755} a^{8} - \frac{74328820906730304543915737833119984045249}{185902597169746701218554007972159810998204} a^{7} + \frac{215783045770913890297482264371494093515721}{464756492924366753046385019930399527495510} a^{6} + \frac{915181926607854969476425367195809594249}{92951298584873350609277003986079905499102} a^{5} - \frac{111505391019320488746963415089354961041853}{464756492924366753046385019930399527495510} a^{4} + \frac{13498463985977548767943082740895802354398}{46475649292436675304638501993039952749551} a^{3} + \frac{73699794748743737237243432746578978528456}{232378246462183376523192509965199763747755} a^{2} - \frac{101997096238977097571293017520868835621506}{232378246462183376523192509965199763747755} a + \frac{102140148356792315054967827088143079028247}{232378246462183376523192509965199763747755}$

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:  $19$
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:  \( 293289608829000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:F_5$ (as 20T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_4:F_5$
Character table for $C_4:F_5$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.260389.1, 5.5.2382032.1, 10.10.8056377164681869568.1

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 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$17$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53Data not computed