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Properties

Label 20.0.78483048970...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $139.55$
Ramified primes $2, 3, 5, 7, 11$
Class number $2989640$ (GRH)
Class group $[2, 1494820]$ (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![16067408281, -4546847540, 7060804611, -1670699804, 1484920771, -295478572, 194491745, -37103844, 18839558, -3322504, 1650224, -363006, 93754, -710, 5128, -2900, 461, 76, -3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 3*x^18 + 76*x^17 + 461*x^16 - 2900*x^15 + 5128*x^14 - 710*x^13 + 93754*x^12 - 363006*x^11 + 1650224*x^10 - 3322504*x^9 + 18839558*x^8 - 37103844*x^7 + 194491745*x^6 - 295478572*x^5 + 1484920771*x^4 - 1670699804*x^3 + 7060804611*x^2 - 4546847540*x + 16067408281)
 
gp: K = bnfinit(x^20 - 6*x^19 - 3*x^18 + 76*x^17 + 461*x^16 - 2900*x^15 + 5128*x^14 - 710*x^13 + 93754*x^12 - 363006*x^11 + 1650224*x^10 - 3322504*x^9 + 18839558*x^8 - 37103844*x^7 + 194491745*x^6 - 295478572*x^5 + 1484920771*x^4 - 1670699804*x^3 + 7060804611*x^2 - 4546847540*x + 16067408281, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 3 x^{18} + 76 x^{17} + 461 x^{16} - 2900 x^{15} + 5128 x^{14} - 710 x^{13} + 93754 x^{12} - 363006 x^{11} + 1650224 x^{10} - 3322504 x^{9} + 18839558 x^{8} - 37103844 x^{7} + 194491745 x^{6} - 295478572 x^{5} + 1484920771 x^{4} - 1670699804 x^{3} + 7060804611 x^{2} - 4546847540 x + 16067408281 \)

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:  \(7848304897073886403551554174576640000000000=2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.55$
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:  $2, 3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4620=2^{2}\cdot 3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4620}(1,·)$, $\chi_{4620}(1219,·)$, $\chi_{4620}(2561,·)$, $\chi_{4620}(841,·)$, $\chi_{4620}(1681,·)$, $\chi_{4620}(4579,·)$, $\chi_{4620}(1301,·)$, $\chi_{4620}(3359,·)$, $\chi_{4620}(419,·)$, $\chi_{4620}(421,·)$, $\chi_{4620}(1259,·)$, $\chi_{4620}(3821,·)$, $\chi_{4620}(2479,·)$, $\chi_{4620}(881,·)$, $\chi_{4620}(839,·)$, $\chi_{4620}(2099,·)$, $\chi_{4620}(1721,·)$, $\chi_{4620}(379,·)$, $\chi_{4620}(2941,·)$, $\chi_{4620}(4159,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{41} a^{15} + \frac{16}{41} a^{14} + \frac{4}{41} a^{13} - \frac{19}{41} a^{12} + \frac{15}{41} a^{11} - \frac{7}{41} a^{10} - \frac{16}{41} a^{9} + \frac{5}{41} a^{8} - \frac{19}{41} a^{7} + \frac{9}{41} a^{6} - \frac{13}{41} a^{5} - \frac{8}{41} a^{4} - \frac{14}{41} a^{3} + \frac{2}{41} a^{2} - \frac{5}{41} a + \frac{3}{41}$, $\frac{1}{1763} a^{16} + \frac{15}{1763} a^{15} + \frac{111}{1763} a^{14} + \frac{428}{1763} a^{13} + \frac{157}{1763} a^{12} - \frac{22}{1763} a^{11} + \frac{32}{1763} a^{10} - \frac{266}{1763} a^{9} + \frac{181}{1763} a^{8} - \frac{792}{1763} a^{7} + \frac{757}{1763} a^{6} - \frac{36}{1763} a^{5} + \frac{35}{1763} a^{4} + \frac{8}{41} a^{3} + \frac{239}{1763} a^{2} - \frac{607}{1763} a - \frac{577}{1763}$, $\frac{1}{1763} a^{17} + \frac{15}{1763} a^{15} + \frac{827}{1763} a^{14} - \frac{458}{1763} a^{13} + \frac{461}{1763} a^{12} + \frac{534}{1763} a^{11} + \frac{114}{1763} a^{10} + \frac{8}{41} a^{9} + \frac{664}{1763} a^{8} - \frac{392}{1763} a^{7} + \frac{348}{1763} a^{6} + \frac{661}{1763} a^{5} + \frac{550}{1763} a^{4} + \frac{325}{1763} a^{3} - \frac{408}{1763} a^{2} + \frac{831}{1763} a + \frac{227}{1763}$, $\frac{1}{69593767956037792961409061} a^{18} + \frac{11611648168935997034754}{69593767956037792961409061} a^{17} - \frac{19706414154006772260470}{69593767956037792961409061} a^{16} + \frac{771160244826064274151232}{69593767956037792961409061} a^{15} - \frac{28767482767619602189984853}{69593767956037792961409061} a^{14} + \frac{24988653537527506295889424}{69593767956037792961409061} a^{13} - \frac{14753221729145837678022413}{69593767956037792961409061} a^{12} + \frac{4942929871466034936914360}{69593767956037792961409061} a^{11} - \frac{21811334905955791938508807}{69593767956037792961409061} a^{10} - \frac{30633738658115344918703663}{69593767956037792961409061} a^{9} + \frac{15444645094742103746162943}{69593767956037792961409061} a^{8} - \frac{10878963317318250565311956}{69593767956037792961409061} a^{7} - \frac{34256839330401320645224193}{69593767956037792961409061} a^{6} + \frac{34387672270997836422585574}{69593767956037792961409061} a^{5} + \frac{300184603994564524814658}{1618459719907855650265327} a^{4} - \frac{18341468339151607615891214}{69593767956037792961409061} a^{3} - \frac{27787962232887537389478835}{69593767956037792961409061} a^{2} + \frac{16130406082470869910897260}{69593767956037792961409061} a - \frac{24876388121011792802672920}{69593767956037792961409061}$, $\frac{1}{14606297475517207145799613056145815080891320327170719516389} a^{19} - \frac{104657150207441680841416810068051}{14606297475517207145799613056145815080891320327170719516389} a^{18} + \frac{2105091628115066511880314709457090052427128974079277294}{14606297475517207145799613056145815080891320327170719516389} a^{17} + \frac{833779046267827696465494669338195001105385186696455775}{14606297475517207145799613056145815080891320327170719516389} a^{16} - \frac{105385176837719134948230678087432017308573711657122960154}{14606297475517207145799613056145815080891320327170719516389} a^{15} + \frac{39432579695251541482000463199944068257985905963732205127}{635056411979009006339113611136774568734405231616118239843} a^{14} + \frac{348298560657326978447640204821777024200183028747487423388}{14606297475517207145799613056145815080891320327170719516389} a^{13} + \frac{3912299669301537452150408644092194999368089556203705683849}{14606297475517207145799613056145815080891320327170719516389} a^{12} + \frac{1468576775002750760827971574832598358535589095394109367689}{14606297475517207145799613056145815080891320327170719516389} a^{11} + \frac{490923630732776288484950269812616546970737254592888151683}{14606297475517207145799613056145815080891320327170719516389} a^{10} + \frac{6120565009761182191270092487531114561789037757521279341141}{14606297475517207145799613056145815080891320327170719516389} a^{9} - \frac{62539635645507871051844060715086097675430245204660488099}{218004439933092643967158403823071866878974930256279395767} a^{8} + \frac{247735121041884465114865531241009686123792868159081499929}{635056411979009006339113611136774568734405231616118239843} a^{7} + \frac{5289729974384361660777982270952002028265966155563092977287}{14606297475517207145799613056145815080891320327170719516389} a^{6} + \frac{42382078281978681025389866971039139306902701784295148081}{14606297475517207145799613056145815080891320327170719516389} a^{5} + \frac{1397216223707677459808868875290587306816293388853729319349}{14606297475517207145799613056145815080891320327170719516389} a^{4} - \frac{4055742635705493553668929281643659360697782957149619722822}{14606297475517207145799613056145815080891320327170719516389} a^{3} - \frac{1365356074769292840817453798271149038803622687024448031670}{14606297475517207145799613056145815080891320327170719516389} a^{2} + \frac{6238863213937390973796185227175543431441875941112937988081}{14606297475517207145799613056145815080891320327170719516389} a - \frac{839260830646446355352787698154308695183391732290030881520}{14606297475517207145799613056145815080891320327170719516389}$

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

Class group and class number

$C_{2}\times C_{1494820}$, which has order $2989640$ (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:  \( 5868059.799558259 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{-5}, \sqrt{21})\), \(\Q(\zeta_{11})^+\), 10.0.685948419200000.1, 10.10.875463320250981.1, 10.0.2801482624803139200000.1

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 R R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$