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Properties

Label 20.0.70952129710...6681.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{18}\cdot 43^{10}$
Root discriminant $98.30$
Ramified primes $3, 11, 43$
Class number $2342400$ (GRH)
Class group $[2, 4, 4, 20, 3660]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121882104697, -49040556030, 80029391166, -29434190142, 24929394483, -8243468934, 4813931163, -1414117766, 634653809, -163790990, 59471534, -13292070, 4000056, -756930, 190108, -29252, 6081, -700, 117, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 117*x^18 - 700*x^17 + 6081*x^16 - 29252*x^15 + 190108*x^14 - 756930*x^13 + 4000056*x^12 - 13292070*x^11 + 59471534*x^10 - 163790990*x^9 + 634653809*x^8 - 1414117766*x^7 + 4813931163*x^6 - 8243468934*x^5 + 24929394483*x^4 - 29434190142*x^3 + 80029391166*x^2 - 49040556030*x + 121882104697)
 
gp: K = bnfinit(x^20 - 8*x^19 + 117*x^18 - 700*x^17 + 6081*x^16 - 29252*x^15 + 190108*x^14 - 756930*x^13 + 4000056*x^12 - 13292070*x^11 + 59471534*x^10 - 163790990*x^9 + 634653809*x^8 - 1414117766*x^7 + 4813931163*x^6 - 8243468934*x^5 + 24929394483*x^4 - 29434190142*x^3 + 80029391166*x^2 - 49040556030*x + 121882104697, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 117 x^{18} - 700 x^{17} + 6081 x^{16} - 29252 x^{15} + 190108 x^{14} - 756930 x^{13} + 4000056 x^{12} - 13292070 x^{11} + 59471534 x^{10} - 163790990 x^{9} + 634653809 x^{8} - 1414117766 x^{7} + 4813931163 x^{6} - 8243468934 x^{5} + 24929394483 x^{4} - 29434190142 x^{3} + 80029391166 x^{2} - 49040556030 x + 121882104697 \)

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:  \(7095212971027263184862290613989426266681=3^{10}\cdot 11^{18}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.30$
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:  $3, 11, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1419=3\cdot 11\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{1419}(128,·)$, $\chi_{1419}(1,·)$, $\chi_{1419}(130,·)$, $\chi_{1419}(388,·)$, $\chi_{1419}(775,·)$, $\chi_{1419}(1289,·)$, $\chi_{1419}(1418,·)$, $\chi_{1419}(1291,·)$, $\chi_{1419}(214,·)$, $\chi_{1419}(644,·)$, $\chi_{1419}(730,·)$, $\chi_{1419}(859,·)$, $\chi_{1419}(988,·)$, $\chi_{1419}(1246,·)$, $\chi_{1419}(1031,·)$, $\chi_{1419}(173,·)$, $\chi_{1419}(431,·)$, $\chi_{1419}(560,·)$, $\chi_{1419}(689,·)$, $\chi_{1419}(1205,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{134} a^{18} + \frac{25}{134} a^{17} - \frac{13}{134} a^{16} - \frac{13}{134} a^{15} - \frac{23}{134} a^{14} + \frac{25}{134} a^{13} - \frac{14}{67} a^{12} + \frac{27}{134} a^{11} - \frac{9}{67} a^{10} - \frac{55}{134} a^{9} + \frac{21}{134} a^{8} - \frac{23}{134} a^{7} + \frac{17}{134} a^{6} - \frac{7}{134} a^{5} - \frac{61}{134} a^{4} - \frac{1}{67} a^{3} - \frac{28}{67} a^{2} - \frac{57}{134} a - \frac{1}{2}$, $\frac{1}{76618706914045744930957378597583630289486003486064034932769215504735118} a^{19} - \frac{269748057641264562258119845335422535331246804036128643047688859943561}{76618706914045744930957378597583630289486003486064034932769215504735118} a^{18} - \frac{12354875872705204025113744535314037893495452059572935000313859827986767}{76618706914045744930957378597583630289486003486064034932769215504735118} a^{17} + \frac{4282232661403458669336858428076808919889893251456096912428313356299239}{76618706914045744930957378597583630289486003486064034932769215504735118} a^{16} + \frac{1447269293016060899128726784976543921644140719909571791862988781468128}{38309353457022872465478689298791815144743001743032017466384607752367559} a^{15} - \frac{6946077572661694291209683960202890631381588149445717986610698001085475}{76618706914045744930957378597583630289486003486064034932769215504735118} a^{14} - \frac{6100449215325558981883855452160721276379217659841309998301633319567724}{38309353457022872465478689298791815144743001743032017466384607752367559} a^{13} - \frac{3035016024977821038233603251994288364722968666480798589430179262529225}{38309353457022872465478689298791815144743001743032017466384607752367559} a^{12} - \frac{6245315092167593751685580896696716441552696549818275646451874217146079}{38309353457022872465478689298791815144743001743032017466384607752367559} a^{11} + \frac{9012573474434474957513936695112992703398878915510651244740142143110037}{38309353457022872465478689298791815144743001743032017466384607752367559} a^{10} + \frac{12562583978460387391606012336593604041903323212740605540761495711940131}{76618706914045744930957378597583630289486003486064034932769215504735118} a^{9} + \frac{18717619809065419924144786637115366301676895573696048606947375073098124}{38309353457022872465478689298791815144743001743032017466384607752367559} a^{8} - \frac{8385093407726256914391806225376501959355343489772592646670487708156621}{76618706914045744930957378597583630289486003486064034932769215504735118} a^{7} - \frac{2199642469696098107357457666503768713906594801312417045623614091493707}{38309353457022872465478689298791815144743001743032017466384607752367559} a^{6} - \frac{11217900987349900398607888902473508763165627691541416814753468918158637}{38309353457022872465478689298791815144743001743032017466384607752367559} a^{5} - \frac{4414952311243846043242980642673794699839992968237058246500926259234225}{38309353457022872465478689298791815144743001743032017466384607752367559} a^{4} - \frac{26464958498002578210843706610626073175644706698456718443757582211592719}{76618706914045744930957378597583630289486003486064034932769215504735118} a^{3} - \frac{3300999652796659963662712512140646728916366662140228267893065514150284}{38309353457022872465478689298791815144743001743032017466384607752367559} a^{2} + \frac{26551827121014478793744539039697744981146673538369557415644655143128649}{76618706914045744930957378597583630289486003486064034932769215504735118} a - \frac{345156192466640448260833271253068450077612352764531298068516776973757}{1143562789761876790014289232799755675962477663971104998996555455294554}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{20}\times C_{3660}$, which has order $2342400$ (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:  \( 125582.779517 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-1419}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{33}, \sqrt{-43})\), \(\Q(\zeta_{11})^+\), 10.0.84233087151233292459.1, \(\Q(\zeta_{33})^+\), 10.0.31512565339032283.3

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

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$