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Properties

Label 20.0.14637626425...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{22}\cdot 3^{18}\cdot 5^{38}\cdot 19^{5}$
Root discriminant $256.02$
Ramified primes $2, 3, 5, 19$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6872217039, 2621446065, -5116531050, -436562865, 710403480, -244662507, 179083080, -74225835, 28987335, -1952580, 609838, 606760, -141585, 10305, -4620, -471, -30, 45, 0, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 45*x^17 - 30*x^16 - 471*x^15 - 4620*x^14 + 10305*x^13 - 141585*x^12 + 606760*x^11 + 609838*x^10 - 1952580*x^9 + 28987335*x^8 - 74225835*x^7 + 179083080*x^6 - 244662507*x^5 + 710403480*x^4 - 436562865*x^3 - 5116531050*x^2 + 2621446065*x + 6872217039)
 
gp: K = bnfinit(x^20 - 5*x^19 + 45*x^17 - 30*x^16 - 471*x^15 - 4620*x^14 + 10305*x^13 - 141585*x^12 + 606760*x^11 + 609838*x^10 - 1952580*x^9 + 28987335*x^8 - 74225835*x^7 + 179083080*x^6 - 244662507*x^5 + 710403480*x^4 - 436562865*x^3 - 5116531050*x^2 + 2621446065*x + 6872217039, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 45 x^{17} - 30 x^{16} - 471 x^{15} - 4620 x^{14} + 10305 x^{13} - 141585 x^{12} + 606760 x^{11} + 609838 x^{10} - 1952580 x^{9} + 28987335 x^{8} - 74225835 x^{7} + 179083080 x^{6} - 244662507 x^{5} + 710403480 x^{4} - 436562865 x^{3} - 5116531050 x^{2} + 2621446065 x + 6872217039 \)

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:  \(1463762642505509948730468750000000000000000000000=2^{22}\cdot 3^{18}\cdot 5^{38}\cdot 19^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $256.02$
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:  $2, 3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{75} a^{10} + \frac{2}{15} a^{9} + \frac{2}{15} a^{7} + \frac{2}{15} a^{6} + \frac{9}{25} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a - \frac{11}{25}$, $\frac{1}{75} a^{11} + \frac{2}{15} a^{8} + \frac{7}{15} a^{7} + \frac{9}{25} a^{6} - \frac{2}{15} a^{4} - \frac{1}{5} a^{2} - \frac{11}{25} a + \frac{2}{5}$, $\frac{1}{300} a^{12} - \frac{1}{300} a^{11} - \frac{1}{20} a^{9} + \frac{67}{300} a^{7} - \frac{13}{75} a^{6} + \frac{23}{60} a^{5} + \frac{1}{30} a^{4} + \frac{9}{20} a^{3} - \frac{3}{50} a^{2} - \frac{29}{100} a - \frac{7}{20}$, $\frac{1}{300} a^{13} - \frac{1}{300} a^{11} + \frac{1}{300} a^{10} + \frac{3}{20} a^{9} - \frac{11}{100} a^{8} - \frac{5}{12} a^{7} + \frac{41}{100} a^{6} - \frac{143}{300} a^{5} - \frac{7}{60} a^{4} + \frac{19}{100} a^{3} - \frac{7}{20} a^{2} - \frac{11}{25} a - \frac{11}{100}$, $\frac{1}{3600} a^{14} - \frac{1}{720} a^{13} - \frac{1}{1200} a^{12} - \frac{1}{900} a^{11} + \frac{1}{450} a^{10} - \frac{29}{300} a^{9} - \frac{19}{180} a^{8} + \frac{587}{1800} a^{7} - \frac{83}{600} a^{6} - \frac{19}{600} a^{5} - \frac{7}{25} a^{4} + \frac{7}{20} a^{3} + \frac{1}{400} a^{2} + \frac{153}{400} a + \frac{119}{400}$, $\frac{1}{3600} a^{15} - \frac{1}{900} a^{13} + \frac{1}{720} a^{12} - \frac{1}{300} a^{11} + \frac{1}{900} a^{10} + \frac{7}{90} a^{9} + \frac{9}{200} a^{8} + \frac{7}{180} a^{7} - \frac{127}{300} a^{6} - \frac{79}{600} a^{5} - \frac{1}{12} a^{4} + \frac{93}{400} a^{3} + \frac{3}{8} a^{2} + \frac{11}{100} a + \frac{131}{400}$, $\frac{1}{14400} a^{16} - \frac{1}{14400} a^{15} - \frac{1}{7200} a^{14} + \frac{23}{14400} a^{13} - \frac{23}{14400} a^{12} - \frac{1}{900} a^{11} - \frac{1}{720} a^{10} - \frac{347}{7200} a^{9} - \frac{187}{7200} a^{8} - \frac{191}{400} a^{7} - \frac{859}{2400} a^{6} + \frac{169}{800} a^{5} + \frac{2387}{4800} a^{4} - \frac{71}{1600} a^{3} - \frac{6}{25} a^{2} + \frac{201}{1600} a + \frac{451}{1600}$, $\frac{1}{28800} a^{17} - \frac{1}{9600} a^{15} - \frac{1}{9600} a^{14} - \frac{1}{1200} a^{13} + \frac{11}{9600} a^{12} - \frac{1}{160} a^{11} - \frac{1}{320} a^{10} + \frac{67}{2400} a^{9} + \frac{1391}{14400} a^{8} - \frac{341}{4800} a^{7} + \frac{11}{30} a^{6} - \frac{4183}{9600} a^{5} - \frac{467}{1600} a^{4} - \frac{1527}{3200} a^{3} - \frac{1087}{3200} a^{2} + \frac{5}{32} a + \frac{1323}{3200}$, $\frac{1}{288000} a^{18} + \frac{1}{96000} a^{17} + \frac{7}{288000} a^{16} - \frac{1}{9600} a^{15} + \frac{11}{288000} a^{14} + \frac{287}{288000} a^{13} + \frac{43}{96000} a^{12} + \frac{149}{144000} a^{11} + \frac{23}{144000} a^{10} + \frac{8711}{144000} a^{9} - \frac{793}{18000} a^{8} - \frac{14563}{48000} a^{7} - \frac{3057}{32000} a^{6} + \frac{103}{32000} a^{5} + \frac{14179}{96000} a^{4} - \frac{677}{3200} a^{3} - \frac{14201}{32000} a^{2} - \frac{10783}{32000} a + \frac{4903}{32000}$, $\frac{1}{1864609497824649505756415265160804711334689932665202456023232000} a^{19} - \frac{165860041176515248010190023670719570736764499685203577153}{233076187228081188219551908145100588916836241583150307002904000} a^{18} - \frac{2956076881107019567729590477468538399198227875203439099}{1472835306338585707548511267899529787784115270667616473952000} a^{17} + \frac{60076097510059785078157707293347766945231941589386837747061}{1864609497824649505756415265160804711334689932665202456023232000} a^{16} - \frac{155202960493689222566160494712456646123131693892297920522199}{1864609497824649505756415265160804711334689932665202456023232000} a^{15} - \frac{922319507478177773029849072640526421681444909346575377907}{37292189956492990115128305303216094226693798653304049120464640} a^{14} + \frac{137648951650370307388084664381645563075079486362298379442279}{93230474891232475287820763258040235566734496633260122801161600} a^{13} + \frac{92402587264345810081314004480336776171353176090568132357083}{74584379912985980230256610606432188453387597306608098240929280} a^{12} + \frac{153755217043376141432655980896993850510702117823436111365493}{23307618722808118821955190814510058891683624158315030700290400} a^{11} - \frac{238234813122945411761290879408674360814690844899427600843753}{93230474891232475287820763258040235566734496633260122801161600} a^{10} - \frac{34051590461561671117654748877356300840979998079282117641074927}{310768249637441584292735877526800785222448322110867076003872000} a^{9} - \frac{3939617441767867847475937045398538491935311574509227576518101}{932304748912324752878207632580402355667344966332601228011616000} a^{8} - \frac{536412148150891949491124860758398368038015978027753159364241867}{1864609497824649505756415265160804711334689932665202456023232000} a^{7} - \frac{46039311763380863392879065711556689709186727717094759553968639}{103589416545813861430911959175600261740816107370289025334624000} a^{6} + \frac{1081889959141647056654764405719167710280619544985403464350991}{38846031204680198036591984690850098152806040263858384500484000} a^{5} - \frac{284380358429393840734803701520432677934273315996546426333782303}{621536499274883168585471755053601570444896644221734152007744000} a^{4} - \frac{89154310122418115074072729900031254896752344768675236077087311}{207178833091627722861823918351200523481632214740578050669248000} a^{3} + \frac{8973459130885086326050304278710105107965078636448621231248773}{25897354136453465357727989793900065435204026842572256333656000} a^{2} - \frac{5470466058217662754980592055454166817185160461110609968166029}{51794708272906930715455979587800130870408053685144512667312000} a + \frac{428141646127451988309336983795409671079735085040167241515429}{981890204225723805032340845266353191856076847111744315968000}$

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:  \( 23307627262679628 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-15}) \), 4.0.17100.2, 5.1.2531250000.8, 10.0.96108398437500000000.105

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.10.19.18$x^{10} + 60$$10$$1$$19$$F_{5}\times C_2$$[9/4]_{4}^{2}$
5.10.19.18$x^{10} + 60$$10$$1$$19$$F_{5}\times C_2$$[9/4]_{4}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$