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Properties

Label 20.0.10946054092...0625.3
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 11^{18}\cdot 17^{10}$
Root discriminant $79.79$
Ramified primes $5, 11, 17$
Class number $357028$ (GRH)
Class group $[2, 178514]$ (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![3800495711, -2552408515, 3643285060, -972550327, 1264916983, -251678261, 344026213, -101632820, 83194308, -26811004, 14106181, -4035680, 1552649, -365324, 108474, -19920, 4609, -606, 107, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 107*x^18 - 606*x^17 + 4609*x^16 - 19920*x^15 + 108474*x^14 - 365324*x^13 + 1552649*x^12 - 4035680*x^11 + 14106181*x^10 - 26811004*x^9 + 83194308*x^8 - 101632820*x^7 + 344026213*x^6 - 251678261*x^5 + 1264916983*x^4 - 972550327*x^3 + 3643285060*x^2 - 2552408515*x + 3800495711)
 
gp: K = bnfinit(x^20 - 8*x^19 + 107*x^18 - 606*x^17 + 4609*x^16 - 19920*x^15 + 108474*x^14 - 365324*x^13 + 1552649*x^12 - 4035680*x^11 + 14106181*x^10 - 26811004*x^9 + 83194308*x^8 - 101632820*x^7 + 344026213*x^6 - 251678261*x^5 + 1264916983*x^4 - 972550327*x^3 + 3643285060*x^2 - 2552408515*x + 3800495711, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 107 x^{18} - 606 x^{17} + 4609 x^{16} - 19920 x^{15} + 108474 x^{14} - 365324 x^{13} + 1552649 x^{12} - 4035680 x^{11} + 14106181 x^{10} - 26811004 x^{9} + 83194308 x^{8} - 101632820 x^{7} + 344026213 x^{6} - 251678261 x^{5} + 1264916983 x^{4} - 972550327 x^{3} + 3643285060 x^{2} - 2552408515 x + 3800495711 \)

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:  \(109460540927745402703534939794619140625=5^{10}\cdot 11^{18}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.79$
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:  $5, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(935=5\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{935}(256,·)$, $\chi_{935}(1,·)$, $\chi_{935}(834,·)$, $\chi_{935}(324,·)$, $\chi_{935}(69,·)$, $\chi_{935}(271,·)$, $\chi_{935}(849,·)$, $\chi_{935}(851,·)$, $\chi_{935}(84,·)$, $\chi_{935}(86,·)$, $\chi_{935}(664,·)$, $\chi_{935}(866,·)$, $\chi_{935}(611,·)$, $\chi_{935}(101,·)$, $\chi_{935}(934,·)$, $\chi_{935}(679,·)$, $\chi_{935}(424,·)$, $\chi_{935}(749,·)$, $\chi_{935}(186,·)$, $\chi_{935}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7589} a^{12} + \frac{2092}{7589} a^{11} + \frac{3692}{7589} a^{10} + \frac{1591}{7589} a^{9} + \frac{3562}{7589} a^{8} + \frac{133}{7589} a^{7} + \frac{2683}{7589} a^{6} - \frac{2886}{7589} a^{5} + \frac{2435}{7589} a^{4} - \frac{2510}{7589} a^{3} + \frac{1745}{7589} a^{2} + \frac{1063}{7589} a - \frac{420}{7589}$, $\frac{1}{7589} a^{13} - \frac{1508}{7589} a^{11} + \frac{3529}{7589} a^{10} - \frac{828}{7589} a^{9} + \frac{827}{7589} a^{8} - \frac{2349}{7589} a^{7} + \frac{138}{7589} a^{6} - \frac{897}{7589} a^{5} + \frac{3278}{7589} a^{4} + \frac{1077}{7589} a^{3} + \frac{832}{7589} a^{2} - \frac{639}{7589} a - \frac{1684}{7589}$, $\frac{1}{7589} a^{14} + \frac{1241}{7589} a^{11} - \frac{3618}{7589} a^{10} + \frac{1931}{7589} a^{9} + \frac{3724}{7589} a^{8} + \frac{3388}{7589} a^{7} + \frac{130}{7589} a^{6} - \frac{313}{7589} a^{5} - \frac{19}{7589} a^{4} + \frac{2663}{7589} a^{3} - \frac{2562}{7589} a^{2} + \frac{41}{7589} a - \frac{3473}{7589}$, $\frac{1}{7589} a^{15} + \frac{3237}{7589} a^{11} - \frac{3674}{7589} a^{10} + \frac{2433}{7589} a^{9} - \frac{256}{7589} a^{8} + \frac{2035}{7589} a^{7} + \frac{1655}{7589} a^{6} - \frac{501}{7589} a^{5} + \frac{1250}{7589} a^{4} + \frac{858}{7589} a^{3} - \frac{2639}{7589} a^{2} - \frac{2170}{7589} a - \frac{2421}{7589}$, $\frac{1}{7589} a^{16} + \frac{1499}{7589} a^{11} - \frac{3485}{7589} a^{10} + \frac{2608}{7589} a^{9} - \frac{468}{7589} a^{8} + \frac{3707}{7589} a^{7} - \frac{3556}{7589} a^{6} + \frac{1173}{7589} a^{5} + \frac{3734}{7589} a^{4} + \frac{2001}{7589} a^{3} + \frac{3070}{7589} a^{2} + \frac{2054}{7589} a + \frac{1109}{7589}$, $\frac{1}{7589} a^{17} + \frac{2453}{7589} a^{11} + \frac{681}{7589} a^{10} - \frac{2431}{7589} a^{9} - \frac{664}{7589} a^{8} + \frac{1980}{7589} a^{7} + \frac{1526}{7589} a^{6} - \frac{3471}{7589} a^{5} + \frac{2245}{7589} a^{4} + \frac{1416}{7589} a^{3} - \frac{3085}{7589} a^{2} + \frac{1362}{7589} a - \frac{307}{7589}$, $\frac{1}{994159} a^{18} - \frac{22}{994159} a^{17} + \frac{37}{994159} a^{16} - \frac{13}{994159} a^{15} - \frac{25}{994159} a^{14} + \frac{16}{994159} a^{13} + \frac{62}{994159} a^{12} + \frac{306340}{994159} a^{11} + \frac{61931}{994159} a^{10} + \frac{463976}{994159} a^{9} - \frac{200576}{994159} a^{8} - \frac{288138}{994159} a^{7} - \frac{299766}{994159} a^{6} - \frac{361677}{994159} a^{5} - \frac{3478}{994159} a^{4} + \frac{137183}{994159} a^{3} - \frac{273029}{994159} a^{2} - \frac{225012}{994159} a - \frac{139180}{994159}$, $\frac{1}{4843655209475330069557738433820602774205366783196125963083} a^{19} - \frac{318249823142825150852128913189720506433624128748608}{4843655209475330069557738433820602774205366783196125963083} a^{18} + \frac{38242211052146953573830420993440618382452631777123047}{4843655209475330069557738433820602774205366783196125963083} a^{17} - \frac{258872886855049805941329794489263632387237476586858226}{4843655209475330069557738433820602774205366783196125963083} a^{16} - \frac{43403486277907674050310176379014852724038112512616738}{4843655209475330069557738433820602774205366783196125963083} a^{15} + \frac{293494892028199361807321683291805807776357478087215897}{4843655209475330069557738433820602774205366783196125963083} a^{14} - \frac{267670199158126738995617531719150864663049346980966585}{4843655209475330069557738433820602774205366783196125963083} a^{13} + \frac{156181831385473671285093606363842457219746966648648119}{4843655209475330069557738433820602774205366783196125963083} a^{12} - \frac{1730455930032654658810439485040585871417117718031610671770}{4843655209475330069557738433820602774205366783196125963083} a^{11} + \frac{1360709442529870512804847704927609793359219606741530432927}{4843655209475330069557738433820602774205366783196125963083} a^{10} - \frac{1350185419539629314253759336003172625225908228291953357108}{4843655209475330069557738433820602774205366783196125963083} a^{9} - \frac{534064248132382576036205806575938774152790628280376564336}{4843655209475330069557738433820602774205366783196125963083} a^{8} + \frac{700729286147343655018028625614919983420738304767744912180}{4843655209475330069557738433820602774205366783196125963083} a^{7} - \frac{278826927523127246991733498229197129177377670604097480501}{4843655209475330069557738433820602774205366783196125963083} a^{6} + \frac{1393150132134783700620204550042466278270218396197797812420}{4843655209475330069557738433820602774205366783196125963083} a^{5} + \frac{1301187280529573981257275449816901430741299222285423300475}{4843655209475330069557738433820602774205366783196125963083} a^{4} + \frac{1961413879860512078443975840835775193394698209374845708592}{4843655209475330069557738433820602774205366783196125963083} a^{3} - \frac{579261671108410228366380615823677799255338518344349186723}{4843655209475330069557738433820602774205366783196125963083} a^{2} + \frac{1866707033318583330443932436991160817900410112376162918978}{4843655209475330069557738433820602774205366783196125963083} a - \frac{7206606336059275179355381984767170402354339994344206017}{54423092241295843478176836335062952519161424530293550147}$

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

Class group and class number

$C_{2}\times C_{178514}$, which has order $357028$ (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:  \( 140644.599182 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-935}) \), \(\Q(\sqrt{-187}) \), \(\Q(\sqrt{5}, \sqrt{-187})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.10462339170938084375.1, 10.0.3347948534700187.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data 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}$
17Data not computed