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Properties

Label 20.0.17930108780...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 3^{18}\cdot 5^{14}\cdot 7^{10}$
Root discriminant $57.90$
Ramified primes $2, 3, 5, 7$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![174972106, 148257512, -110117546, -74437464, 32556156, 5720946, -2028876, -45090, 140118, 2264, -83405, 36236, 5145, -9318, 4596, -1644, 486, -156, 49, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 49*x^18 - 156*x^17 + 486*x^16 - 1644*x^15 + 4596*x^14 - 9318*x^13 + 5145*x^12 + 36236*x^11 - 83405*x^10 + 2264*x^9 + 140118*x^8 - 45090*x^7 - 2028876*x^6 + 5720946*x^5 + 32556156*x^4 - 74437464*x^3 - 110117546*x^2 + 148257512*x + 174972106)
 
gp: K = bnfinit(x^20 - 10*x^19 + 49*x^18 - 156*x^17 + 486*x^16 - 1644*x^15 + 4596*x^14 - 9318*x^13 + 5145*x^12 + 36236*x^11 - 83405*x^10 + 2264*x^9 + 140118*x^8 - 45090*x^7 - 2028876*x^6 + 5720946*x^5 + 32556156*x^4 - 74437464*x^3 - 110117546*x^2 + 148257512*x + 174972106, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 49 x^{18} - 156 x^{17} + 486 x^{16} - 1644 x^{15} + 4596 x^{14} - 9318 x^{13} + 5145 x^{12} + 36236 x^{11} - 83405 x^{10} + 2264 x^{9} + 140118 x^{8} - 45090 x^{7} - 2028876 x^{6} + 5720946 x^{5} + 32556156 x^{4} - 74437464 x^{3} - 110117546 x^{2} + 148257512 x + 174972106 \)

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:  \(179301087802125125222400000000000000=2^{28}\cdot 3^{18}\cdot 5^{14}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.90$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{105} a^{12} - \frac{2}{35} a^{11} - \frac{10}{21} a^{9} + \frac{1}{5} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{6} - \frac{11}{35} a^{5} - \frac{17}{35} a^{4} + \frac{7}{15} a^{3} + \frac{11}{35} a^{2} + \frac{1}{5} a + \frac{46}{105}$, $\frac{1}{105} a^{13} - \frac{1}{105} a^{11} - \frac{1}{7} a^{10} + \frac{12}{35} a^{9} - \frac{13}{35} a^{8} + \frac{2}{7} a^{7} - \frac{1}{35} a^{6} - \frac{13}{35} a^{5} - \frac{47}{105} a^{4} + \frac{4}{35} a^{3} + \frac{44}{105} a^{2} - \frac{1}{35} a - \frac{13}{35}$, $\frac{1}{105} a^{14} + \frac{2}{15} a^{11} + \frac{1}{105} a^{10} + \frac{17}{35} a^{9} + \frac{17}{35} a^{8} + \frac{2}{5} a^{7} + \frac{12}{35} a^{6} + \frac{5}{21} a^{5} - \frac{13}{35} a^{4} - \frac{4}{35} a^{3} - \frac{8}{21} a^{2} + \frac{52}{105} a - \frac{8}{35}$, $\frac{1}{105} a^{15} + \frac{1}{7} a^{11} + \frac{16}{105} a^{10} + \frac{17}{35} a^{9} - \frac{2}{5} a^{8} + \frac{12}{35} a^{7} + \frac{5}{21} a^{6} + \frac{1}{35} a^{5} - \frac{11}{35} a^{4} + \frac{3}{35} a^{3} + \frac{3}{7} a^{2} - \frac{38}{105} a + \frac{1}{5}$, $\frac{1}{525} a^{16} + \frac{2}{525} a^{15} - \frac{1}{525} a^{12} + \frac{37}{525} a^{11} + \frac{13}{525} a^{10} + \frac{32}{105} a^{9} - \frac{23}{175} a^{8} + \frac{16}{75} a^{7} + \frac{113}{525} a^{6} - \frac{43}{175} a^{5} + \frac{8}{175} a^{4} + \frac{32}{75} a^{3} - \frac{8}{75} a^{2} - \frac{41}{525} a - \frac{134}{525}$, $\frac{1}{525} a^{17} + \frac{1}{525} a^{15} - \frac{1}{525} a^{13} - \frac{1}{525} a^{12} + \frac{79}{525} a^{11} + \frac{13}{175} a^{10} - \frac{78}{175} a^{9} + \frac{10}{21} a^{8} - \frac{52}{175} a^{7} - \frac{16}{105} a^{6} + \frac{2}{25} a^{5} - \frac{7}{75} a^{4} + \frac{206}{525} a^{3} - \frac{149}{525} a^{2} - \frac{69}{175} a + \frac{36}{175}$, $\frac{1}{159777045231594508209732547125} a^{18} - \frac{3}{53259015077198169403244182375} a^{17} - \frac{15613169052328862105877353}{22825292175942072601390363875} a^{16} - \frac{647348678132388562163558953}{159777045231594508209732547125} a^{15} - \frac{28327437076683514230483107}{12290541940891885246902503625} a^{14} - \frac{42278441292434334464947699}{12290541940891885246902503625} a^{13} - \frac{30747248741920915896447681}{10651803015439633880648836475} a^{12} + \frac{4391398743006969812935792533}{53259015077198169403244182375} a^{11} + \frac{12057192177955106525046966614}{159777045231594508209732547125} a^{10} - \frac{12263582920307070291821096524}{159777045231594508209732547125} a^{9} + \frac{4186258467241822655574574734}{53259015077198169403244182375} a^{8} + \frac{2872692157867127728928358191}{31955409046318901641946509425} a^{7} + \frac{36719922008630312407351059596}{159777045231594508209732547125} a^{6} - \frac{29408173182848033881309575284}{159777045231594508209732547125} a^{5} - \frac{6516705007881462300000520033}{22825292175942072601390363875} a^{4} - \frac{25236790718841803906596251947}{53259015077198169403244182375} a^{3} - \frac{14828789457011898753876639963}{53259015077198169403244182375} a^{2} + \frac{14229580936248710833885720178}{159777045231594508209732547125} a - \frac{20029564154797868215137762959}{159777045231594508209732547125}$, $\frac{1}{2395940995750596700910766073191709875} a^{19} + \frac{7497754}{2395940995750596700910766073191709875} a^{18} + \frac{473792668924742147178862007764529}{798646998583532233636922024397236625} a^{17} - \frac{165768573772498660160947734103114}{266215666194510744545640674799078875} a^{16} - \frac{15987679711907012494462416807373}{4095625633761703762240625766139675} a^{15} - \frac{35435127208154663917653478081381}{12286876901285111286721877298419025} a^{14} + \frac{360982879710453916073692527452181}{266215666194510744545640674799078875} a^{13} - \frac{712984801122856636936250319144744}{266215666194510744545640674799078875} a^{12} + \frac{73464620473963988052165722843320567}{798646998583532233636922024397236625} a^{11} - \frac{311998048404679103485151564627576767}{2395940995750596700910766073191709875} a^{10} + \frac{70909108278147329536966765713160753}{479188199150119340182153214638341975} a^{9} - \frac{17328323224937211933327524774765314}{114092428369076033376703146342462375} a^{8} - \frac{13516226509382875622174678799687403}{38030809456358677792234382114154125} a^{7} - \frac{129573231595174843436047144329413704}{266215666194510744545640674799078875} a^{6} - \frac{203028528341654389299635802023019341}{798646998583532233636922024397236625} a^{5} + \frac{77973239417626646799032531873485984}{266215666194510744545640674799078875} a^{4} - \frac{11484255936137488313831742216795344}{38030809456358677792234382114154125} a^{3} + \frac{152876202254409065222810442163983007}{798646998583532233636922024397236625} a^{2} + \frac{156398706681850061666953906076709041}{479188199150119340182153214638341975} a + \frac{21779171003797513294400026805353666}{184303153519276669300828159476285375}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 2178006063.441225 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{7}, \sqrt{-15})\), 5.1.162000.1, 10.0.393660000000.1, 10.2.28229306112000000.1, 10.0.423439591680000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

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 ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$