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Properties

Label 20.0.10675123826...000.12
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{34}\cdot 7^{10}$
Root discriminant $282.76$
Ramified primes $2, 3, 5, 7$
Class number $741548032$ (GRH)
Class group $[2, 4, 4, 4, 4, 4, 362084]$ (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![907898452801, 67569922460, 353939485660, 18422918540, 66433466785, 3111251244, 7909442290, 361124040, 654022215, 27516620, 39756432, 1333680, 1829830, 29080, 64390, -44, 1805, -20, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 40*x^18 - 20*x^17 + 1805*x^16 - 44*x^15 + 64390*x^14 + 29080*x^13 + 1829830*x^12 + 1333680*x^11 + 39756432*x^10 + 27516620*x^9 + 654022215*x^8 + 361124040*x^7 + 7909442290*x^6 + 3111251244*x^5 + 66433466785*x^4 + 18422918540*x^3 + 353939485660*x^2 + 67569922460*x + 907898452801)
 
gp: K = bnfinit(x^20 + 40*x^18 - 20*x^17 + 1805*x^16 - 44*x^15 + 64390*x^14 + 29080*x^13 + 1829830*x^12 + 1333680*x^11 + 39756432*x^10 + 27516620*x^9 + 654022215*x^8 + 361124040*x^7 + 7909442290*x^6 + 3111251244*x^5 + 66433466785*x^4 + 18422918540*x^3 + 353939485660*x^2 + 67569922460*x + 907898452801, 1)
 

Normalized defining polynomial

\( x^{20} + 40 x^{18} - 20 x^{17} + 1805 x^{16} - 44 x^{15} + 64390 x^{14} + 29080 x^{13} + 1829830 x^{12} + 1333680 x^{11} + 39756432 x^{10} + 27516620 x^{9} + 654022215 x^{8} + 361124040 x^{7} + 7909442290 x^{6} + 3111251244 x^{5} + 66433466785 x^{4} + 18422918540 x^{3} + 353939485660 x^{2} + 67569922460 x + 907898452801 \)

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:  \(10675123826048640000000000000000000000000000000000=2^{40}\cdot 3^{10}\cdot 5^{34}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $282.76$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4200=2^{3}\cdot 3\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{4200}(1,·)$, $\chi_{4200}(1091,·)$, $\chi_{4200}(839,·)$, $\chi_{4200}(841,·)$, $\chi_{4200}(1931,·)$, $\chi_{4200}(589,·)$, $\chi_{4200}(1679,·)$, $\chi_{4200}(1681,·)$, $\chi_{4200}(2771,·)$, $\chi_{4200}(1429,·)$, $\chi_{4200}(2519,·)$, $\chi_{4200}(2521,·)$, $\chi_{4200}(3611,·)$, $\chi_{4200}(2269,·)$, $\chi_{4200}(3359,·)$, $\chi_{4200}(3361,·)$, $\chi_{4200}(3109,·)$, $\chi_{4200}(4199,·)$, $\chi_{4200}(3949,·)$, $\chi_{4200}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{25} a^{10} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{3}{25} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{25}$, $\frac{1}{25} a^{11} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{3}{25} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{25} a$, $\frac{1}{25} a^{12} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{3}{25} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{25} a^{2} - \frac{1}{5}$, $\frac{1}{25} a^{13} - \frac{2}{5} a^{9} + \frac{3}{25} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{4} - \frac{1}{25} a^{3} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{25} a^{14} + \frac{3}{25} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{25} a^{4} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{325} a^{15} - \frac{2}{325} a^{13} - \frac{2}{325} a^{12} - \frac{6}{325} a^{11} + \frac{1}{65} a^{10} - \frac{2}{65} a^{9} - \frac{41}{325} a^{8} - \frac{106}{325} a^{7} + \frac{157}{325} a^{6} - \frac{16}{65} a^{5} - \frac{29}{65} a^{4} + \frac{122}{325} a^{3} - \frac{38}{325} a^{2} + \frac{96}{325} a + \frac{128}{325}$, $\frac{1}{2275} a^{16} + \frac{2}{2275} a^{15} + \frac{37}{2275} a^{14} - \frac{9}{455} a^{13} + \frac{3}{2275} a^{12} + \frac{19}{2275} a^{11} + \frac{1}{175} a^{10} + \frac{836}{2275} a^{9} - \frac{191}{455} a^{8} - \frac{731}{2275} a^{7} - \frac{51}{175} a^{6} + \frac{839}{2275} a^{5} - \frac{141}{325} a^{4} - \frac{172}{455} a^{3} + \frac{131}{325} a^{2} + \frac{684}{2275} a + \frac{828}{2275}$, $\frac{1}{2275} a^{17} - \frac{2}{2275} a^{15} - \frac{4}{325} a^{14} - \frac{19}{2275} a^{13} - \frac{8}{2275} a^{12} + \frac{3}{2275} a^{11} - \frac{2}{2275} a^{10} - \frac{639}{2275} a^{9} + \frac{703}{2275} a^{8} - \frac{769}{2275} a^{7} - \frac{691}{2275} a^{6} - \frac{411}{2275} a^{5} + \frac{638}{2275} a^{4} - \frac{996}{2275} a^{3} - \frac{639}{2275} a^{2} - \frac{41}{175} a + \frac{32}{175}$, $\frac{1}{12012834121574658319675} a^{18} - \frac{243036940970300089}{2402566824314931663935} a^{17} - \frac{391826149302077796}{2402566824314931663935} a^{16} + \frac{10474512615087585926}{12012834121574658319675} a^{15} + \frac{195627558719335255294}{12012834121574658319675} a^{14} + \frac{29181629784843935613}{1716119160224951188525} a^{13} - \frac{25836855903838576041}{12012834121574658319675} a^{12} - \frac{8113192765484754331}{2402566824314931663935} a^{11} - \frac{87349575829241901079}{12012834121574658319675} a^{10} - \frac{3027985771002380616643}{12012834121574658319675} a^{9} + \frac{1288547629816011849483}{12012834121574658319675} a^{8} + \frac{287271402203034524679}{924064163198050639975} a^{7} - \frac{131324997927332992357}{343223832044990237705} a^{6} + \frac{2948996401740982233023}{12012834121574658319675} a^{5} + \frac{4814473912474555821246}{12012834121574658319675} a^{4} + \frac{4335531581819013554252}{12012834121574658319675} a^{3} - \frac{1754827326932392813194}{12012834121574658319675} a^{2} + \frac{207735738053378138286}{2402566824314931663935} a - \frac{5210224770746480556}{279368235385457170225}$, $\frac{1}{1488312669256819431126574919108207676960915314313979075} a^{19} + \frac{38742154902092951159231366082308}{1488312669256819431126574919108207676960915314313979075} a^{18} - \frac{148312100179108437100677772199139473850327723017742}{1488312669256819431126574919108207676960915314313979075} a^{17} - \frac{143802901160480379898164904258308464078376076130981}{1488312669256819431126574919108207676960915314313979075} a^{16} - \frac{2135945131257923929645071761648491133028960266238119}{1488312669256819431126574919108207676960915314313979075} a^{15} - \frac{1549662374624538241475928296257450444055232919928728}{212616095608117061589510702729743953851559330616282725} a^{14} - \frac{11791541954325472077259904382595949965611818924988572}{1488312669256819431126574919108207676960915314313979075} a^{13} - \frac{303115240227966854783370124471550891918443178615118}{114485589942832263932813455316015975150839639562613775} a^{12} + \frac{16105348193207290511324671664395606294577100260116358}{1488312669256819431126574919108207676960915314313979075} a^{11} + \frac{17204442253462348735133739572449762961430480847425883}{1488312669256819431126574919108207676960915314313979075} a^{10} - \frac{391493249689887272238631182350573017177092414304294988}{1488312669256819431126574919108207676960915314313979075} a^{9} + \frac{522209177556991141766501902572808428229028420629832954}{1488312669256819431126574919108207676960915314313979075} a^{8} + \frac{43669563048947455466657359427981299216718527192313134}{212616095608117061589510702729743953851559330616282725} a^{7} - \frac{473945600133738145776244795216108651533273450908854281}{1488312669256819431126574919108207676960915314313979075} a^{6} + \frac{410394650939773266970744282180336261622422977193698799}{1488312669256819431126574919108207676960915314313979075} a^{5} + \frac{619374219172831705100701956182436747123660471465481819}{1488312669256819431126574919108207676960915314313979075} a^{4} + \frac{62207263603028904377818296464001298228452412181687251}{1488312669256819431126574919108207676960915314313979075} a^{3} - \frac{389497020534768977523422735329706902325437497699582547}{1488312669256819431126574919108207676960915314313979075} a^{2} - \frac{525729258134749290503455870064222258478985565912039446}{1488312669256819431126574919108207676960915314313979075} a - \frac{296988605762840540351564878029754625262842714283}{899829001452132219944180556233971491425860002185}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{362084}$, which has order $741548032$ (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:  \( 19344397.966990974 \) (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

\(\Q(\sqrt{-42}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{10}, \sqrt{-42})\), 5.5.390625.1, 10.0.20420505000000000000000.1, 10.10.25000000000000000.1, 10.0.3190703906250000000000.3

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 R R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
5Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$