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Properties

Label 20.0.73038164166...9296.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $49.34$
Ramified primes $2, 11, 23$
Class number $160$ (GRH)
Class group $[2, 4, 20]$ (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2935789, 3465188, 3930518, 2976592, 997121, 977364, 567360, 41876, 48918, 56748, 30356, -2204, -3650, 2908, 544, 252, 125, 0, 22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 22*x^18 + 125*x^16 + 252*x^15 + 544*x^14 + 2908*x^13 - 3650*x^12 - 2204*x^11 + 30356*x^10 + 56748*x^9 + 48918*x^8 + 41876*x^7 + 567360*x^6 + 977364*x^5 + 997121*x^4 + 2976592*x^3 + 3930518*x^2 + 3465188*x + 2935789)
 
gp: K = bnfinit(x^20 - 4*x^19 + 22*x^18 + 125*x^16 + 252*x^15 + 544*x^14 + 2908*x^13 - 3650*x^12 - 2204*x^11 + 30356*x^10 + 56748*x^9 + 48918*x^8 + 41876*x^7 + 567360*x^6 + 977364*x^5 + 997121*x^4 + 2976592*x^3 + 3930518*x^2 + 3465188*x + 2935789, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 22 x^{18} + 125 x^{16} + 252 x^{15} + 544 x^{14} + 2908 x^{13} - 3650 x^{12} - 2204 x^{11} + 30356 x^{10} + 56748 x^{9} + 48918 x^{8} + 41876 x^{7} + 567360 x^{6} + 977364 x^{5} + 997121 x^{4} + 2976592 x^{3} + 3930518 x^{2} + 3465188 x + 2935789 \)

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:  \(7303816416639774784171798530359296=2^{30}\cdot 11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.34$
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:  $2, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} + \frac{1}{8} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{11} - \frac{1}{8} a^{7} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{8} + \frac{1}{16}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{9} + \frac{1}{16} a$, $\frac{1}{5296} a^{18} - \frac{79}{5296} a^{17} + \frac{1}{5296} a^{16} - \frac{15}{2648} a^{15} + \frac{35}{2648} a^{14} - \frac{31}{2648} a^{13} + \frac{19}{1324} a^{12} - \frac{113}{2648} a^{11} - \frac{22}{331} a^{10} + \frac{75}{1324} a^{9} + \frac{161}{2648} a^{8} - \frac{437}{2648} a^{7} - \frac{83}{2648} a^{6} - \frac{589}{2648} a^{5} + \frac{81}{1324} a^{4} + \frac{437}{2648} a^{3} - \frac{609}{5296} a^{2} + \frac{795}{5296} a - \frac{1171}{5296}$, $\frac{1}{242861702513797860218558409612464089500208487907863024} a^{19} + \frac{1391119909462610003585165793141575929166069955425}{15178856407112366263659900600779005593763030494241439} a^{18} + \frac{4651621510186146594950124151115722903531377972051707}{242861702513797860218558409612464089500208487907863024} a^{17} - \frac{362892907014199596244678795463315411860546783784673}{30357712814224732527319801201558011187526060988482878} a^{16} + \frac{2727760644545681926818349424081489057502423599914125}{121430851256898930109279204806232044750104243953931512} a^{15} - \frac{5710910116725037860569836585861237862187907106915439}{121430851256898930109279204806232044750104243953931512} a^{14} - \frac{207697840829455522945293713403049052975706612062071}{5279602228560823048229530643749219336961054084953544} a^{13} - \frac{1024538576383226823936150079178913783848106162651189}{60715425628449465054639602403116022375052121976965756} a^{12} + \frac{23125978411003454316058569843152469642687800549415}{15178856407112366263659900600779005593763030494241439} a^{11} - \frac{2655404711968244929873374550456633835900000097002853}{121430851256898930109279204806232044750104243953931512} a^{10} - \frac{7538929782886980912086562361054376861426933183151663}{60715425628449465054639602403116022375052121976965756} a^{9} - \frac{1433939552270797860310418170552627955966790144553097}{30357712814224732527319801201558011187526060988482878} a^{8} - \frac{12619981985161441940657193869887832666727512338237421}{121430851256898930109279204806232044750104243953931512} a^{7} + \frac{24720781907916435787447132795863948148961879038422447}{121430851256898930109279204806232044750104243953931512} a^{6} + \frac{10547278726387059344449845132570992665438369526516669}{121430851256898930109279204806232044750104243953931512} a^{5} - \frac{5037302574098517192960119090714803954654330860614807}{60715425628449465054639602403116022375052121976965756} a^{4} + \frac{107032361014932167888413159850295536924081772861771471}{242861702513797860218558409612464089500208487907863024} a^{3} + \frac{27737464708390925492120217089965940799075606438394077}{121430851256898930109279204806232044750104243953931512} a^{2} - \frac{32962126886738953647961541298047807744733946593358103}{242861702513797860218558409612464089500208487907863024} a - \frac{329395728046441356351000950765547854716804861787117}{659950278570102881028691330468652417120131760619193}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{20}$, which has order $160$ (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:  \( 2417625.83378 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.5048580365312.1, 10.0.2670699013250048.2, 10.10.116117348402176.1

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 32 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$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}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$