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Properties

Label 20.0.42700495304...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{32}\cdot 7^{10}$
Root discriminant $240.73$
Ramified primes $2, 3, 5, 7$
Class number $219351040$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 214210]$ (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![363424022857, 52695486900, 169841406000, 19586049980, 36987800005, 3305115036, 5007659530, 326939720, 473428615, 20522700, 33249096, 812240, 1793970, 15920, 75350, -204, 2445, -20, 60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 60*x^18 - 20*x^17 + 2445*x^16 - 204*x^15 + 75350*x^14 + 15920*x^13 + 1793970*x^12 + 812240*x^11 + 33249096*x^10 + 20522700*x^9 + 473428615*x^8 + 326939720*x^7 + 5007659530*x^6 + 3305115036*x^5 + 36987800005*x^4 + 19586049980*x^3 + 169841406000*x^2 + 52695486900*x + 363424022857)
 
gp: K = bnfinit(x^20 + 60*x^18 - 20*x^17 + 2445*x^16 - 204*x^15 + 75350*x^14 + 15920*x^13 + 1793970*x^12 + 812240*x^11 + 33249096*x^10 + 20522700*x^9 + 473428615*x^8 + 326939720*x^7 + 5007659530*x^6 + 3305115036*x^5 + 36987800005*x^4 + 19586049980*x^3 + 169841406000*x^2 + 52695486900*x + 363424022857, 1)
 

Normalized defining polynomial

\( x^{20} + 60 x^{18} - 20 x^{17} + 2445 x^{16} - 204 x^{15} + 75350 x^{14} + 15920 x^{13} + 1793970 x^{12} + 812240 x^{11} + 33249096 x^{10} + 20522700 x^{9} + 473428615 x^{8} + 326939720 x^{7} + 5007659530 x^{6} + 3305115036 x^{5} + 36987800005 x^{4} + 19586049980 x^{3} + 169841406000 x^{2} + 52695486900 x + 363424022857 \)

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:  \(427004953041945600000000000000000000000000000000=2^{40}\cdot 3^{10}\cdot 5^{32}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $240.73$
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$$\chi_{4200}(1,·)$, $\chi_{4200}(1091,·)$, $\chi_{4200}(3781,·)$, $\chi_{4200}(841,·)$, $\chi_{4200}(1931,·)$, $\chi_{4200}(1681,·)$, $\chi_{4200}(2771,·)$, $\chi_{4200}(2521,·)$, $\chi_{4200}(3611,·)$, $\chi_{4200}(671,·)$, $\chi_{4200}(3361,·)$, $\chi_{4200}(421,·)$, $\chi_{4200}(1511,·)$, $\chi_{4200}(1261,·)$, $\chi_{4200}(2351,·)$, $\chi_{4200}(2101,·)$, $\chi_{4200}(3191,·)$, $\chi_{4200}(251,·)$, $\chi_{4200}(2941,·)$, $\chi_{4200}(4031,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} + \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{7} a^{13} + \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{77} a^{15} + \frac{4}{77} a^{13} - \frac{4}{77} a^{12} + \frac{30}{77} a^{10} - \frac{38}{77} a^{9} + \frac{17}{77} a^{8} - \frac{23}{77} a^{7} + \frac{3}{11} a^{6} - \frac{20}{77} a^{5} - \frac{25}{77} a^{4} + \frac{20}{77} a^{3} - \frac{25}{77} a^{2} - \frac{1}{11} a + \frac{4}{11}$, $\frac{1}{539} a^{16} - \frac{2}{539} a^{15} - \frac{29}{539} a^{14} - \frac{1}{539} a^{13} + \frac{8}{539} a^{12} - \frac{24}{77} a^{11} + \frac{155}{539} a^{10} + \frac{258}{539} a^{9} + \frac{185}{539} a^{8} + \frac{67}{539} a^{7} - \frac{23}{77} a^{6} - \frac{128}{539} a^{5} + \frac{37}{539} a^{4} + \frac{177}{539} a^{3} + \frac{25}{77} a^{2} + \frac{17}{77} a + \frac{2}{11}$, $\frac{1}{539} a^{17} + \frac{2}{539} a^{15} + \frac{18}{539} a^{14} - \frac{8}{539} a^{13} + \frac{16}{539} a^{12} - \frac{258}{539} a^{11} + \frac{155}{539} a^{10} + \frac{141}{539} a^{9} - \frac{123}{539} a^{8} - \frac{139}{539} a^{7} - \frac{23}{539} a^{6} + \frac{236}{539} a^{5} - \frac{239}{539} a^{4} - \frac{80}{539} a^{3} + \frac{19}{77} a^{2} + \frac{13}{77} a + \frac{2}{11}$, $\frac{1}{844861979074417907029273} a^{18} - \frac{98559884117969355238}{844861979074417907029273} a^{17} - \frac{81831254809931832983}{844861979074417907029273} a^{16} - \frac{494139667042248613112}{76805634461310718820843} a^{15} + \frac{20211964856405339506640}{844861979074417907029273} a^{14} - \frac{34186577389096863293058}{844861979074417907029273} a^{13} - \frac{457005671523452812832}{844861979074417907029273} a^{12} + \frac{116324491824840026649528}{844861979074417907029273} a^{11} - \frac{289960521993612714432536}{844861979074417907029273} a^{10} - \frac{349943307841540035453156}{844861979074417907029273} a^{9} - \frac{177284455455273809620106}{844861979074417907029273} a^{8} - \frac{420930748725478970554292}{844861979074417907029273} a^{7} + \frac{16164935192916144007381}{76805634461310718820843} a^{6} - \frac{364553884782564932021969}{844861979074417907029273} a^{5} + \frac{8586621804716312784366}{76805634461310718820843} a^{4} + \frac{378470915886897784754835}{844861979074417907029273} a^{3} + \frac{25863786663855979801689}{120694568439202558147039} a^{2} - \frac{49266868869869845352255}{120694568439202558147039} a - \frac{4168795148841758841147}{17242081205600365449577}$, $\frac{1}{454415734883445072416486174461936728799942101794685124777} a^{19} - \frac{82188207972727195000192790817582}{454415734883445072416486174461936728799942101794685124777} a^{18} + \frac{6384002820847042010723947417849626497778402951698246}{41310521353040461128771470405630611709085645617698647707} a^{17} + \frac{256542473267311806799523216508995807693297805073461833}{454415734883445072416486174461936728799942101794685124777} a^{16} - \frac{36692620896559904961703580115826648768767262878700003}{9273790507825409641152779070651769975509022485605818873} a^{15} + \frac{1414576336854605965424651143309723745623168805697221675}{64916533554777867488069453494562389828563157399240732111} a^{14} + \frac{8103867439065043117134570940060329712306796227738730082}{454415734883445072416486174461936728799942101794685124777} a^{13} + \frac{30987769649969821851199653337629120310730397467349802828}{454415734883445072416486174461936728799942101794685124777} a^{12} - \frac{1183926478049503705751119769923611001952818244324689338}{9273790507825409641152779070651769975509022485605818873} a^{11} + \frac{129628045963825265161469182137852502659292027828303490342}{454415734883445072416486174461936728799942101794685124777} a^{10} - \frac{178948614804382568978810648014446521934080186650045511822}{454415734883445072416486174461936728799942101794685124777} a^{9} - \frac{143442250770396070190993729483286583583625935523841286298}{454415734883445072416486174461936728799942101794685124777} a^{8} - \frac{99131763868354473465200600249725026366363532806898167565}{454415734883445072416486174461936728799942101794685124777} a^{7} + \frac{56208008987165195232223502696200274824416170774198101981}{454415734883445072416486174461936728799942101794685124777} a^{6} - \frac{137214771386990383279324129862465821111243146396799115227}{454415734883445072416486174461936728799942101794685124777} a^{5} - \frac{186771498007459923546466064705006132948775484516394809632}{454415734883445072416486174461936728799942101794685124777} a^{4} + \frac{74212169207482096271109442804608529650284165371352682689}{454415734883445072416486174461936728799942101794685124777} a^{3} - \frac{14897231407962501806812949635515477702416493452932501590}{64916533554777867488069453494562389828563157399240732111} a^{2} + \frac{32012801373196311138803192624186153255902181257653386328}{64916533554777867488069453494562389828563157399240732111} a + \frac{1685598130234764532574541690893489761472058190738901120}{9273790507825409641152779070651769975509022485605818873}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{214210}$, which has order $219351040$ (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:  \( 42294001.73672045 \) (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

\(\Q(\sqrt{-42}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{2}, \sqrt{-21})\), 5.5.390625.1, 10.0.20420505000000000000000.1, 10.10.5000000000000000.1, 10.0.638140781250000000000.3

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 ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$5$5.10.16.7$x^{10} + 40 x^{9} + 10 x^{8} + 70 x^{7} + 15 x^{6} + 95 x^{5} + 5 x^{4} + 80 x^{3} + 5 x^{2} + 90 x + 7$$5$$2$$16$$C_{10}$$[2]^{2}$
5.10.16.7$x^{10} + 40 x^{9} + 10 x^{8} + 70 x^{7} + 15 x^{6} + 95 x^{5} + 5 x^{4} + 80 x^{3} + 5 x^{2} + 90 x + 7$$5$$2$$16$$C_{10}$$[2]^{2}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$