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Properties

Label 20.0.16863534401...000.13
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $204.81$
Ramified primes $2, 5, 7, 11$
Class number $48499392$ (GRH)
Class group $[2, 2, 12124848]$ (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![110525629055689, 0, 5656755393990, 0, 1253186524445, 0, 111162014200, 0, 9387080850, 0, 521554516, 0, 21746530, 0, 618840, 0, 12325, 0, 150, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 150*x^18 + 12325*x^16 + 618840*x^14 + 21746530*x^12 + 521554516*x^10 + 9387080850*x^8 + 111162014200*x^6 + 1253186524445*x^4 + 5656755393990*x^2 + 110525629055689)
 
gp: K = bnfinit(x^20 + 150*x^18 + 12325*x^16 + 618840*x^14 + 21746530*x^12 + 521554516*x^10 + 9387080850*x^8 + 111162014200*x^6 + 1253186524445*x^4 + 5656755393990*x^2 + 110525629055689, 1)
 

Normalized defining polynomial

\( x^{20} + 150 x^{18} + 12325 x^{16} + 618840 x^{14} + 21746530 x^{12} + 521554516 x^{10} + 9387080850 x^{8} + 111162014200 x^{6} + 1253186524445 x^{4} + 5656755393990 x^{2} + 110525629055689 \)

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:  \(16863534401027144382603387201975746560000000000=2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $204.81$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3080=2^{3}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3080}(1,·)$, $\chi_{3080}(1539,·)$, $\chi_{3080}(699,·)$, $\chi_{3080}(841,·)$, $\chi_{3080}(139,·)$, $\chi_{3080}(2829,·)$, $\chi_{3080}(589,·)$, $\chi_{3080}(1681,·)$, $\chi_{3080}(1429,·)$, $\chi_{3080}(2071,·)$, $\chi_{3080}(29,·)$, $\chi_{3080}(2659,·)$, $\chi_{3080}(1511,·)$, $\chi_{3080}(1961,·)$, $\chi_{3080}(111,·)$, $\chi_{3080}(2549,·)$, $\chi_{3080}(951,·)$, $\chi_{3080}(1401,·)$, $\chi_{3080}(2939,·)$, $\chi_{3080}(1791,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{4} - \frac{1}{5} a^{2} - \frac{1}{20}$, $\frac{1}{20} a^{5} - \frac{1}{5} a^{3} - \frac{1}{20} a$, $\frac{1}{40} a^{6} - \frac{1}{40} a^{4} + \frac{7}{40} a^{2} + \frac{17}{40}$, $\frac{1}{40} a^{7} - \frac{1}{40} a^{5} + \frac{7}{40} a^{3} + \frac{17}{40} a$, $\frac{1}{400} a^{8} + \frac{1}{200} a^{6} + \frac{1}{100} a^{4} + \frac{39}{200} a^{2} + \frac{171}{400}$, $\frac{1}{400} a^{9} + \frac{1}{200} a^{7} + \frac{1}{100} a^{5} + \frac{39}{200} a^{3} + \frac{171}{400} a$, $\frac{1}{800} a^{10} - \frac{1}{800} a^{8} - \frac{1}{400} a^{6} - \frac{7}{400} a^{4} - \frac{143}{800} a^{2} - \frac{33}{800}$, $\frac{1}{1600} a^{11} - \frac{1}{1600} a^{10} - \frac{1}{1600} a^{9} + \frac{1}{1600} a^{8} + \frac{9}{800} a^{7} - \frac{9}{800} a^{6} - \frac{17}{800} a^{5} + \frac{17}{800} a^{4} + \frac{397}{1600} a^{3} - \frac{397}{1600} a^{2} - \frac{93}{1600} a + \frac{93}{1600}$, $\frac{1}{8000} a^{12} - \frac{1}{4000} a^{10} - \frac{1}{1600} a^{8} + \frac{3}{400} a^{6} - \frac{29}{1600} a^{4} - \frac{381}{4000} a^{2} - \frac{171}{8000}$, $\frac{1}{16000} a^{13} - \frac{1}{16000} a^{12} - \frac{1}{8000} a^{11} + \frac{1}{8000} a^{10} - \frac{1}{3200} a^{9} + \frac{1}{3200} a^{8} + \frac{3}{800} a^{7} - \frac{3}{800} a^{6} - \frac{29}{3200} a^{5} + \frac{29}{3200} a^{4} + \frac{1619}{8000} a^{3} - \frac{1619}{8000} a^{2} - \frac{4171}{16000} a + \frac{4171}{16000}$, $\frac{1}{16000} a^{14} - \frac{1}{16000} a^{12} - \frac{7}{16000} a^{10} + \frac{3}{3200} a^{8} - \frac{33}{3200} a^{6} - \frac{267}{16000} a^{4} + \frac{747}{16000} a^{2} + \frac{189}{16000}$, $\frac{1}{32000} a^{15} - \frac{1}{32000} a^{14} - \frac{1}{32000} a^{13} + \frac{1}{32000} a^{12} - \frac{7}{32000} a^{11} + \frac{7}{32000} a^{10} - \frac{1}{1280} a^{9} + \frac{1}{1280} a^{8} - \frac{49}{6400} a^{7} + \frac{49}{6400} a^{6} - \frac{427}{32000} a^{5} + \frac{427}{32000} a^{4} + \frac{5627}{32000} a^{3} - \frac{5627}{32000} a^{2} - \frac{14651}{32000} a + \frac{14651}{32000}$, $\frac{1}{160000} a^{16} + \frac{1}{40000} a^{14} - \frac{1}{20000} a^{12} + \frac{3}{40000} a^{10} + \frac{17}{16000} a^{8} - \frac{273}{40000} a^{6} + \frac{61}{5000} a^{4} - \frac{9971}{40000} a^{2} - \frac{9539}{160000}$, $\frac{1}{320000} a^{17} - \frac{1}{320000} a^{16} + \frac{1}{80000} a^{15} - \frac{1}{80000} a^{14} - \frac{1}{40000} a^{13} + \frac{1}{40000} a^{12} + \frac{3}{80000} a^{11} - \frac{3}{80000} a^{10} + \frac{17}{32000} a^{9} - \frac{17}{32000} a^{8} + \frac{727}{80000} a^{7} - \frac{727}{80000} a^{6} - \frac{4}{625} a^{5} + \frac{4}{625} a^{4} + \frac{17029}{80000} a^{3} - \frac{17029}{80000} a^{2} - \frac{21539}{320000} a + \frac{21539}{320000}$, $\frac{1}{92835336148329140747135782579520000} a^{18} + \frac{49058146126192182224779827431}{92835336148329140747135782579520000} a^{16} - \frac{45192509650732028776281160153}{4641766807416457037356789128976000} a^{14} + \frac{128629502686295105266614756559}{23208834037082285186783945644880000} a^{12} + \frac{5338803084674168996949559270827}{46417668074164570373567891289760000} a^{10} + \frac{3494377462862975517592727354049}{46417668074164570373567891289760000} a^{8} + \frac{15257013819562014155377542022667}{23208834037082285186783945644880000} a^{6} + \frac{67411498008845110836017501134127}{4641766807416457037356789128976000} a^{4} - \frac{7384568605187442334910748233747087}{92835336148329140747135782579520000} a^{2} + \frac{26704438372989617991460999759396527}{92835336148329140747135782579520000}$, $\frac{1}{1951977501323427222368211794290111127680000} a^{19} - \frac{1}{185670672296658281494271565159040000} a^{18} - \frac{1127662075822844040606331082233985179}{1951977501323427222368211794290111127680000} a^{17} + \frac{531162704800864947444818813691}{185670672296658281494271565159040000} a^{16} + \frac{96171377160213208913305154596191041}{24399718766542840279602647428626389096000} a^{15} - \frac{322184364068462775311496080459}{23208834037082285186783945644880000} a^{14} - \frac{1664667732035592580465837881192206209}{121998593832714201398013237143131945480000} a^{13} + \frac{80740461388616729784092282001}{23208834037082285186783945644880000} a^{12} - \frac{16226018713689054356609394797070945283}{975988750661713611184105897145055563840000} a^{11} - \frac{1582873331574821585898235163881}{3713413445933165629885431303180800} a^{10} - \frac{800394236585545215105271437962904151551}{975988750661713611184105897145055563840000} a^{9} + \frac{60329916139113308746063123169371}{92835336148329140747135782579520000} a^{8} + \frac{2838985318248333960898620124838420025011}{243997187665428402796026474286263890960000} a^{7} + \frac{30926469994367042100122833131513}{11604417018541142593391972822440000} a^{6} + \frac{509436602727914004878809666615032374519}{48799437533085680559205294857252778192000} a^{5} - \frac{105224850750796327354336127328191}{11604417018541142593391972822440000} a^{4} - \frac{213706303770190197249191907180551550585827}{1951977501323427222368211794290111127680000} a^{3} + \frac{15070173996567241074514251834049099}{185670672296658281494271565159040000} a^{2} - \frac{836261607035968242688513955024816371903983}{1951977501323427222368211794290111127680000} a + \frac{13594155618725844182771416022224267}{37134134459331656298854313031808000}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{12124848}$, which has order $48499392$ (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:  \( 8013735.512306594 \) (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

\(\Q(\sqrt{-770}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{7}, \sqrt{-110})\), \(\Q(\zeta_{11})^+\), 10.0.4058114748686028800000.1, 10.10.3689195226078208.1, 10.0.241453843558400000.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed