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Properties

Label 20.4.45653946346...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 61^{10}$
Root discriminant $30.41$
Ramified primes $2, 5, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\wr C_2$ (as 20T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18749, 346539, -168442, -107895, -65918, 149377, -20328, -60333, 56319, -27034, 3064, 6738, -5479, 2049, -408, 11, 66, -67, 34, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 34*x^18 - 67*x^17 + 66*x^16 + 11*x^15 - 408*x^14 + 2049*x^13 - 5479*x^12 + 6738*x^11 + 3064*x^10 - 27034*x^9 + 56319*x^8 - 60333*x^7 - 20328*x^6 + 149377*x^5 - 65918*x^4 - 107895*x^3 - 168442*x^2 + 346539*x - 18749)
 
gp: K = bnfinit(x^20 - 9*x^19 + 34*x^18 - 67*x^17 + 66*x^16 + 11*x^15 - 408*x^14 + 2049*x^13 - 5479*x^12 + 6738*x^11 + 3064*x^10 - 27034*x^9 + 56319*x^8 - 60333*x^7 - 20328*x^6 + 149377*x^5 - 65918*x^4 - 107895*x^3 - 168442*x^2 + 346539*x - 18749, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 34 x^{18} - 67 x^{17} + 66 x^{16} + 11 x^{15} - 408 x^{14} + 2049 x^{13} - 5479 x^{12} + 6738 x^{11} + 3064 x^{10} - 27034 x^{9} + 56319 x^{8} - 60333 x^{7} - 20328 x^{6} + 149377 x^{5} - 65918 x^{4} - 107895 x^{3} - 168442 x^{2} + 346539 x - 18749 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(456539463464244864640000000000=2^{16}\cdot 5^{10}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.41$
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:  $2, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{16} a^{16} - \frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{3}{16} a^{12} - \frac{3}{8} a^{11} - \frac{1}{8} a^{10} - \frac{3}{8} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} - \frac{1}{4} a^{5} + \frac{7}{16} a^{4} + \frac{7}{16} a^{3} - \frac{1}{2} a^{2} + \frac{7}{16} a - \frac{1}{16}$, $\frac{1}{16} a^{17} - \frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{4} a^{13} - \frac{1}{16} a^{12} - \frac{1}{2} a^{10} + \frac{3}{8} a^{9} - \frac{1}{8} a^{7} + \frac{3}{8} a^{6} - \frac{5}{16} a^{5} - \frac{1}{8} a^{4} + \frac{7}{16} a^{3} - \frac{1}{16} a^{2} - \frac{1}{8} a + \frac{7}{16}$, $\frac{1}{32} a^{18} - \frac{1}{32} a^{17} - \frac{1}{32} a^{16} - \frac{3}{32} a^{14} - \frac{5}{32} a^{13} + \frac{1}{32} a^{12} - \frac{1}{2} a^{11} - \frac{5}{16} a^{10} - \frac{7}{16} a^{9} + \frac{3}{16} a^{8} - \frac{1}{2} a^{7} + \frac{13}{32} a^{6} - \frac{5}{32} a^{5} + \frac{1}{32} a^{4} + \frac{7}{32} a^{2} - \frac{7}{32} a - \frac{15}{32}$, $\frac{1}{127776651782698451689488268582866989600581723552} a^{19} + \frac{178886309295460021899971784754151179048987405}{127776651782698451689488268582866989600581723552} a^{18} - \frac{1407190520829112059148369143102952267521865673}{127776651782698451689488268582866989600581723552} a^{17} + \frac{249452747051297860795632070382729384145676539}{63888325891349225844744134291433494800290861776} a^{16} - \frac{7031453620336255192486854563334150550909136045}{127776651782698451689488268582866989600581723552} a^{15} - \frac{14966555938470506149484907774959171179814273093}{127776651782698451689488268582866989600581723552} a^{14} - \frac{18019646740957947018027454166153491323837913873}{127776651782698451689488268582866989600581723552} a^{13} + \frac{7430391021196567342287994563462997887685661167}{31944162945674612922372067145716747400145430888} a^{12} + \frac{9549175173549189807513360714360697919663978303}{63888325891349225844744134291433494800290861776} a^{11} - \frac{6953203592899500545429886461839787408834360249}{63888325891349225844744134291433494800290861776} a^{10} - \frac{6200281195722003906830721580738401507258954857}{63888325891349225844744134291433494800290861776} a^{9} - \frac{7797691550672626844880136454887782873664972683}{31944162945674612922372067145716747400145430888} a^{8} + \frac{41162650814003826080421133156048935314385616305}{127776651782698451689488268582866989600581723552} a^{7} - \frac{49796451367922368416236749079942013525215300867}{127776651782698451689488268582866989600581723552} a^{6} - \frac{40529543695910693728186726194300234831928285907}{127776651782698451689488268582866989600581723552} a^{5} - \frac{6500639175401361145189595141907858324007300817}{63888325891349225844744134291433494800290861776} a^{4} - \frac{62549852734714348018187174845904057726496057651}{127776651782698451689488268582866989600581723552} a^{3} + \frac{41319560407935023519540977665059991196201658245}{127776651782698451689488268582866989600581723552} a^{2} - \frac{44008622630852829166975142070497376358090583121}{127776651782698451689488268582866989600581723552} a - \frac{11338342638225151671920694418255729890751249079}{31944162945674612922372067145716747400145430888}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 9876199.95752 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{305}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{61})\), 10.2.2976800000.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$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}$
61Data not computed