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Properties

Label 20.0.379...597.1
Degree $20$
Signature $[0, 10]$
Discriminant $3.794\times 10^{20}$
Root discriminant \(10.69\)
Ramified primes $3,31$
Class number $1$
Class group trivial
Galois group $C_{10}\wr C_2$ (as 20T53)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^20 - 5*y^19 + 6*y^18 + 14*y^17 - 46*y^16 + 25*y^15 + 73*y^14 - 122*y^13 - 8*y^12 + 193*y^11 - 146*y^10 - 144*y^9 + 308*y^8 - 127*y^7 - 128*y^6 + 161*y^5 - 46*y^4 - 26*y^3 + 25*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1)
 

\( x^{20} - 5 x^{19} + 6 x^{18} + 14 x^{17} - 46 x^{16} + 25 x^{15} + 73 x^{14} - 122 x^{13} - 8 x^{12} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(379379679498607236597\) \(\medspace = 3^{15}\cdot 31^{9}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(10.69\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{3/4}31^{9/10}\approx 50.12639424180317$
Ramified primes:   \(3\), \(31\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{93}) \)
$\card{ \Aut(K/\Q) }$:  $10$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6978527}a^{19}-\frac{502850}{6978527}a^{18}+\frac{2639465}{6978527}a^{17}-\frac{2706308}{6978527}a^{16}-\frac{1189948}{6978527}a^{15}-\frac{438476}{6978527}a^{14}-\frac{1096272}{6978527}a^{13}+\frac{110407}{6978527}a^{12}-\frac{3425638}{6978527}a^{11}-\frac{707323}{6978527}a^{10}-\frac{751820}{6978527}a^{9}+\frac{1184585}{6978527}a^{8}+\frac{3485122}{6978527}a^{7}-\frac{557869}{6978527}a^{6}-\frac{1191169}{6978527}a^{5}-\frac{574971}{6978527}a^{4}+\frac{918839}{6978527}a^{3}+\frac{718635}{6978527}a^{2}+\frac{68564}{6978527}a-\frac{3141208}{6978527}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{294846}{18413} a^{19} - \frac{1356362}{18413} a^{18} + \frac{1211607}{18413} a^{17} + \frac{4680751}{18413} a^{16} - \frac{11744765}{18413} a^{15} + \frac{2419699}{18413} a^{14} + \frac{23070963}{18413} a^{13} - \frac{26770987}{18413} a^{12} - \frac{14299518}{18413} a^{11} + \frac{52373757}{18413} a^{10} - \frac{21096924}{18413} a^{9} - \frac{53479978}{18413} a^{8} + \frac{69996438}{18413} a^{7} - \frac{6459642}{18413} a^{6} - \frac{43411115}{18413} a^{5} + \frac{29757726}{18413} a^{4} + \frac{526458}{18413} a^{3} - \frac{8417968}{18413} a^{2} + \frac{3705327}{18413} a - \frac{536855}{18413} \)  (order $6$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{125833687}{6978527}a^{19}-\frac{579002682}{6978527}a^{18}+\frac{515643741}{6978527}a^{17}+\frac{2002422506}{6978527}a^{16}-\frac{5008590138}{6978527}a^{15}+\frac{1006953784}{6978527}a^{14}+\frac{9854695436}{6978527}a^{13}-\frac{11369476744}{6978527}a^{12}-\frac{6154055041}{6978527}a^{11}+\frac{22289927428}{6978527}a^{10}-\frac{8890563197}{6978527}a^{9}-\frac{22816363911}{6978527}a^{8}+\frac{29713303298}{6978527}a^{7}-\frac{2673076729}{6978527}a^{6}-\frac{18404064846}{6978527}a^{5}+\frac{12598467425}{6978527}a^{4}+\frac{154360922}{6978527}a^{3}-\frac{3559981387}{6978527}a^{2}+\frac{1601390146}{6978527}a-\frac{243752067}{6978527}$, $\frac{26762083}{6978527}a^{19}-\frac{117368506}{6978527}a^{18}+\frac{86709828}{6978527}a^{17}+\frac{433441327}{6978527}a^{16}-\frac{961078852}{6978527}a^{15}+\frac{43486694}{6978527}a^{14}+\frac{2010621278}{6978527}a^{13}-\frac{1961957160}{6978527}a^{12}-\frac{1555419814}{6978527}a^{11}+\frac{4200168647}{6978527}a^{10}-\frac{1092157736}{6978527}a^{9}-\frac{4696192865}{6978527}a^{8}+\frac{5151441259}{6978527}a^{7}+\frac{145118254}{6978527}a^{6}-\frac{3371710596}{6978527}a^{5}+\frac{1898278360}{6978527}a^{4}+\frac{172875141}{6978527}a^{3}-\frac{553195917}{6978527}a^{2}+\frac{232796404}{6978527}a-\frac{41793447}{6978527}$, $\frac{32889832}{6978527}a^{19}-\frac{108291887}{6978527}a^{18}-\frac{38389679}{6978527}a^{17}+\frac{600949327}{6978527}a^{16}-\frac{584549267}{6978527}a^{15}-\frac{1029611448}{6978527}a^{14}+\frac{2196029046}{6978527}a^{13}+\frac{144684768}{6978527}a^{12}-\frac{3717504884}{6978527}a^{11}+\frac{2590635940}{6978527}a^{10}+\frac{3396101846}{6978527}a^{9}-\frac{5806702029}{6978527}a^{8}+\frac{182070987}{6978527}a^{7}+\frac{5133858817}{6978527}a^{6}-\frac{2556534395}{6978527}a^{5}-\frac{1606071875}{6978527}a^{4}+\frac{1603082447}{6978527}a^{3}-\frac{46882479}{6978527}a^{2}-\frac{289828720}{6978527}a+\frac{85694051}{6978527}$, $\frac{49888150}{6978527}a^{19}-\frac{196294250}{6978527}a^{18}+\frac{67284737}{6978527}a^{17}+\frac{862418800}{6978527}a^{16}-\frac{1415541011}{6978527}a^{15}-\frac{654298751}{6978527}a^{14}+\frac{3635118793}{6978527}a^{13}-\frac{1978758124}{6978527}a^{12}-\frac{4211634325}{6978527}a^{11}+\frac{6229334093}{6978527}a^{10}+\frac{1194244276}{6978527}a^{9}-\frac{8989844543}{6978527}a^{8}+\frac{5549404629}{6978527}a^{7}+\frac{3779580930}{6978527}a^{6}-\frac{5377213314}{6978527}a^{5}+\frac{842212256}{6978527}a^{4}+\frac{1302476091}{6978527}a^{3}-\frac{609409967}{6978527}a^{2}+\frac{41000185}{6978527}a+\frac{13681465}{6978527}$, $\frac{21911531}{6978527}a^{19}-\frac{68310022}{6978527}a^{18}-\frac{46654449}{6978527}a^{17}+\frac{423339453}{6978527}a^{16}-\frac{312213610}{6978527}a^{15}-\frac{893649489}{6978527}a^{14}+\frac{1485656329}{6978527}a^{13}+\frac{554558403}{6978527}a^{12}-\frac{2923629334}{6978527}a^{11}+\frac{1315972066}{6978527}a^{10}+\frac{3256528943}{6978527}a^{9}-\frac{4035533311}{6978527}a^{8}-\frac{1089980004}{6978527}a^{7}+\frac{4527532959}{6978527}a^{6}-\frac{1322406858}{6978527}a^{5}-\frac{2051278318}{6978527}a^{4}+\frac{1284251518}{6978527}a^{3}+\frac{205263506}{6978527}a^{2}-\frac{302136264}{6978527}a+\frac{60791716}{6978527}$, $\frac{5463093}{6978527}a^{19}-\frac{47075608}{6978527}a^{18}+\frac{114802428}{6978527}a^{17}+\frac{34582096}{6978527}a^{16}-\frac{591786798}{6978527}a^{15}+\frac{767692668}{6978527}a^{14}+\frac{564049534}{6978527}a^{13}-\frac{2181637237}{6978527}a^{12}+\frac{999990953}{6978527}a^{11}+\frac{2637459188}{6978527}a^{10}-\frac{3602583653}{6978527}a^{9}-\frac{876415285}{6978527}a^{8}+\frac{5424789617}{6978527}a^{7}-\frac{3510202350}{6978527}a^{6}-\frac{1930628723}{6978527}a^{5}+\frac{3313500046}{6978527}a^{4}-\frac{865749110}{6978527}a^{3}-\frac{658884616}{6978527}a^{2}+\frac{445697455}{6978527}a-\frac{80874670}{6978527}$, $\frac{90156599}{6978527}a^{19}-\frac{424522713}{6978527}a^{18}+\frac{410667618}{6978527}a^{17}+\frac{1408085930}{6978527}a^{16}-\frac{3748926881}{6978527}a^{15}+\frac{1054999798}{6978527}a^{14}+\frac{7080240169}{6978527}a^{13}-\frac{8890703433}{6978527}a^{12}-\frac{3750741737}{6978527}a^{11}+\frac{16622331053}{6978527}a^{10}-\frac{7894453200}{6978527}a^{9}-\frac{16086644095}{6978527}a^{8}+\frac{23104946010}{6978527}a^{7}-\frac{3693359184}{6978527}a^{6}-\frac{13478916000}{6978527}a^{5}+\frac{10394821692}{6978527}a^{4}-\frac{517448015}{6978527}a^{3}-\frac{2746485796}{6978527}a^{2}+\frac{1383327906}{6978527}a-\frac{231596543}{6978527}$, $\frac{61136830}{6978527}a^{19}-\frac{261628832}{6978527}a^{18}+\frac{170092465}{6978527}a^{17}+\frac{1009733892}{6978527}a^{16}-\frac{2091572718}{6978527}a^{15}-\frac{123807846}{6978527}a^{14}+\frac{4593747192}{6978527}a^{13}-\frac{4010054276}{6978527}a^{12}-\frac{3989204187}{6978527}a^{11}+\frac{9208005573}{6978527}a^{10}-\frac{1533184215}{6978527}a^{9}-\frac{10935043913}{6978527}a^{8}+\frac{10666902952}{6978527}a^{7}+\frac{1473681256}{6978527}a^{6}-\frac{7624310740}{6978527}a^{5}+\frac{3576937525}{6978527}a^{4}+\frac{776114034}{6978527}a^{3}-\frac{1202818371}{6978527}a^{2}+\frac{405532123}{6978527}a-\frac{47186348}{6978527}$, $\frac{18668047}{6978527}a^{19}-\frac{68818427}{6978527}a^{18}+\frac{15181078}{6978527}a^{17}+\frac{300999886}{6978527}a^{16}-\frac{453520573}{6978527}a^{15}-\frac{236687532}{6978527}a^{14}+\frac{1143451952}{6978527}a^{13}-\frac{597123535}{6978527}a^{12}-\frac{1269009233}{6978527}a^{11}+\frac{1888293362}{6978527}a^{10}+\frac{300106238}{6978527}a^{9}-\frac{2626797553}{6978527}a^{8}+\frac{1784397631}{6978527}a^{7}+\frac{749995199}{6978527}a^{6}-\frac{1404416450}{6978527}a^{5}+\frac{589181007}{6978527}a^{4}+\frac{101068945}{6978527}a^{3}-\frac{217704765}{6978527}a^{2}+\frac{115058289}{6978527}a-\frac{19083180}{6978527}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 181.753135766 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 181.753135766 \cdot 1}{6\cdot\sqrt{379379679498607236597}}\cr\approx \mathstrut & 0.149139373015 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_{10}\wr C_2$ (as 20T53):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_{10}\wr C_2$
Character table for $C_{10}\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.837.1, 10.0.224415603.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.4.0.1}{4} }^{5}$ ${\href{/padicField/7.5.0.1}{5} }^{4}$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ R ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}$ $20$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ $20$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ $20$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.20.15.1$x^{20} + 23 x^{16} + 4 x^{15} - 606 x^{12} - 756 x^{11} + 6 x^{10} + 1862 x^{8} + 7788 x^{7} + 1044 x^{6} + 4 x^{5} + 11581 x^{4} - 5788 x^{3} + 894 x^{2} - 52 x + 4$$4$$5$$15$20T12$[\ ]_{4}^{10}$
\(31\) Copy content Toggle raw display 31.10.9.3$x^{10} + 527$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.0.1$x^{10} + 30 x^{5} + 26 x^{4} + 13 x^{3} + 13 x^{2} + 13 x + 3$$1$$10$$0$$C_{10}$$[\ ]^{10}$