↑

Properties

Label 20.0.82300358279...5056.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{18}\cdot 13^{10}$
Root discriminant $88.26$
Ramified primes $2, 11, 13$
Class number $140052$ (GRH)
Class group $[140052]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![328199719, 60230034, -30848325, -57625378, 27205499, -1607844, 6672306, -5844692, 2425991, -1026202, 626493, -284654, 102753, -36380, 13682, -4140, 1053, -246, 59, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 59*x^18 - 246*x^17 + 1053*x^16 - 4140*x^15 + 13682*x^14 - 36380*x^13 + 102753*x^12 - 284654*x^11 + 626493*x^10 - 1026202*x^9 + 2425991*x^8 - 5844692*x^7 + 6672306*x^6 - 1607844*x^5 + 27205499*x^4 - 57625378*x^3 - 30848325*x^2 + 60230034*x + 328199719)
 
gp: K = bnfinit(x^20 - 10*x^19 + 59*x^18 - 246*x^17 + 1053*x^16 - 4140*x^15 + 13682*x^14 - 36380*x^13 + 102753*x^12 - 284654*x^11 + 626493*x^10 - 1026202*x^9 + 2425991*x^8 - 5844692*x^7 + 6672306*x^6 - 1607844*x^5 + 27205499*x^4 - 57625378*x^3 - 30848325*x^2 + 60230034*x + 328199719, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 59 x^{18} - 246 x^{17} + 1053 x^{16} - 4140 x^{15} + 13682 x^{14} - 36380 x^{13} + 102753 x^{12} - 284654 x^{11} + 626493 x^{10} - 1026202 x^{9} + 2425991 x^{8} - 5844692 x^{7} + 6672306 x^{6} - 1607844 x^{5} + 27205499 x^{4} - 57625378 x^{3} - 30848325 x^{2} + 60230034 x + 328199719 \)

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:  \(823003582791542802072855307752419885056=2^{30}\cdot 11^{18}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.26$
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:  $2, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1144=2^{3}\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1144}(1,·)$, $\chi_{1144}(261,·)$, $\chi_{1144}(441,·)$, $\chi_{1144}(961,·)$, $\chi_{1144}(521,·)$, $\chi_{1144}(909,·)$, $\chi_{1144}(1117,·)$, $\chi_{1144}(469,·)$, $\chi_{1144}(25,·)$, $\chi_{1144}(729,·)$, $\chi_{1144}(285,·)$, $\chi_{1144}(805,·)$, $\chi_{1144}(677,·)$, $\chi_{1144}(753,·)$, $\chi_{1144}(1065,·)$, $\chi_{1144}(365,·)$, $\chi_{1144}(989,·)$, $\chi_{1144}(625,·)$, $\chi_{1144}(313,·)$, $\chi_{1144}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{8} a^{3} - \frac{1}{8} a$, $\frac{1}{184} a^{14} - \frac{7}{184} a^{13} - \frac{1}{184} a^{11} + \frac{11}{184} a^{10} + \frac{1}{92} a^{9} - \frac{1}{184} a^{8} - \frac{9}{184} a^{7} - \frac{25}{184} a^{6} - \frac{9}{92} a^{5} - \frac{1}{184} a^{4} - \frac{5}{184} a^{3} - \frac{27}{92} a^{2} + \frac{61}{184} a - \frac{3}{8}$, $\frac{1}{184} a^{15} - \frac{3}{184} a^{13} - \frac{1}{184} a^{12} + \frac{1}{46} a^{11} - \frac{13}{184} a^{10} + \frac{13}{184} a^{9} - \frac{2}{23} a^{8} + \frac{1}{46} a^{7} - \frac{9}{184} a^{6} - \frac{35}{184} a^{5} - \frac{3}{46} a^{4} + \frac{3}{184} a^{3} - \frac{41}{184} a^{2} - \frac{7}{23} a - \frac{1}{8}$, $\frac{1}{368} a^{16} - \frac{11}{184} a^{13} + \frac{1}{92} a^{12} + \frac{15}{184} a^{11} - \frac{1}{8} a^{10} + \frac{9}{92} a^{9} - \frac{45}{368} a^{8} - \frac{41}{184} a^{7} + \frac{37}{184} a^{6} - \frac{5}{92} a^{5} - \frac{1}{8} a^{4} - \frac{5}{184} a^{3} - \frac{63}{184} a^{2} + \frac{57}{184} a + \frac{1}{16}$, $\frac{1}{15824} a^{17} + \frac{13}{15824} a^{16} - \frac{3}{7912} a^{15} + \frac{21}{7912} a^{13} + \frac{17}{989} a^{12} - \frac{771}{7912} a^{11} - \frac{351}{7912} a^{10} + \frac{39}{368} a^{9} - \frac{1053}{15824} a^{8} - \frac{1619}{7912} a^{7} - \frac{1663}{7912} a^{6} + \frac{429}{3956} a^{5} + \frac{1055}{7912} a^{4} + \frac{665}{1978} a^{3} - \frac{1923}{7912} a^{2} + \frac{1435}{15824} a - \frac{83}{688}$, $\frac{1}{3672411598358692698750224} a^{18} - \frac{9}{3672411598358692698750224} a^{17} - \frac{4878506504669127392605}{3672411598358692698750224} a^{16} - \frac{111183167013878991541}{459051449794836587343778} a^{15} - \frac{1078581644701349111147}{918102899589673174687556} a^{14} - \frac{36993311343405659646487}{1836205799179346349375112} a^{13} - \frac{21868503129091269948559}{459051449794836587343778} a^{12} - \frac{920745352190461683233}{39917517373464051073372} a^{11} - \frac{320281664753473078801561}{3672411598358692698750224} a^{10} - \frac{13506616645606016394865}{159670069493856204293488} a^{9} + \frac{13007264930344278689461}{159670069493856204293488} a^{8} + \frac{2492440494827205803203}{918102899589673174687556} a^{7} - \frac{149133029906108188186797}{1836205799179346349375112} a^{6} - \frac{434027077356688738521359}{1836205799179346349375112} a^{5} + \frac{38880351597745821996461}{459051449794836587343778} a^{4} - \frac{9717602370503799053463}{39917517373464051073372} a^{3} - \frac{651821732310982152326309}{3672411598358692698750224} a^{2} + \frac{1689486713522282912979647}{3672411598358692698750224} a - \frac{57064913562604241134203}{159670069493856204293488}$, $\frac{1}{1624671218702287291234400347376} a^{19} + \frac{110595}{812335609351143645617200173688} a^{18} + \frac{14184174369636136792563643}{1624671218702287291234400347376} a^{17} + \frac{503872519670712266958422623}{812335609351143645617200173688} a^{16} + \frac{191349636142477596496711167}{812335609351143645617200173688} a^{15} - \frac{1444440719240338271291733925}{812335609351143645617200173688} a^{14} - \frac{29141927088077341256518083485}{812335609351143645617200173688} a^{13} - \frac{1109520395642632916970137147}{35318939537006245461617398856} a^{12} + \frac{186680457300658838272543157043}{1624671218702287291234400347376} a^{11} + \frac{83608655512175252349463097007}{812335609351143645617200173688} a^{10} - \frac{2787261130291028705873494577}{70637879074012490923234797712} a^{9} - \frac{8628000959449438924029799991}{203083902337785911404300043422} a^{8} + \frac{136569947156933779922141350869}{812335609351143645617200173688} a^{7} - \frac{7358367576705620628183574043}{101541951168892955702150021711} a^{6} - \frac{178072880145683220078506301901}{812335609351143645617200173688} a^{5} + \frac{194909511566824609801669295861}{812335609351143645617200173688} a^{4} - \frac{531679580146723464448440851411}{1624671218702287291234400347376} a^{3} + \frac{28545956081909625135779362713}{203083902337785911404300043422} a^{2} + \frac{744357966280980338980989196435}{1624671218702287291234400347376} a - \frac{8837194132474434291418463117}{35318939537006245461617398856}$

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

Class group and class number

$C_{140052}$, which has order $140052$ (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:  \( 2015201.7242 \) (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

\(\Q(\sqrt{-286}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{13}, \sqrt{-22})\), \(\Q(\zeta_{11})^+\), 10.0.28688039019625283584.1, 10.10.79589952003133.1, 10.0.77265229938688.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
11Data not computed
$13$13.10.5.1$x^{10} - 676 x^{6} + 114244 x^{2} - 13366548$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
13.10.5.1$x^{10} - 676 x^{6} + 114244 x^{2} - 13366548$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$