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Properties

Label 20.0.16706271954...6304.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{18}\cdot 23^{4}$
Root discriminant $45.83$
Ramified primes $2, 11, 23$
Class number $48$ (GRH)
Class group $[2, 2, 12]$ (GRH)
Galois group $C_2^2\times C_2^4:C_5$ (as 20T74)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![145993, 111368, -17626, -289242, -201043, -18128, 101736, 97580, 99859, 61954, 21262, 11260, 6945, 668, -268, 44, 14, 12, 10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 10*x^18 + 12*x^17 + 14*x^16 + 44*x^15 - 268*x^14 + 668*x^13 + 6945*x^12 + 11260*x^11 + 21262*x^10 + 61954*x^9 + 99859*x^8 + 97580*x^7 + 101736*x^6 - 18128*x^5 - 201043*x^4 - 289242*x^3 - 17626*x^2 + 111368*x + 145993)
 
gp: K = bnfinit(x^20 - 2*x^19 + 10*x^18 + 12*x^17 + 14*x^16 + 44*x^15 - 268*x^14 + 668*x^13 + 6945*x^12 + 11260*x^11 + 21262*x^10 + 61954*x^9 + 99859*x^8 + 97580*x^7 + 101736*x^6 - 18128*x^5 - 201043*x^4 - 289242*x^3 - 17626*x^2 + 111368*x + 145993, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 10 x^{18} + 12 x^{17} + 14 x^{16} + 44 x^{15} - 268 x^{14} + 668 x^{13} + 6945 x^{12} + 11260 x^{11} + 21262 x^{10} + 61954 x^{9} + 99859 x^{8} + 97580 x^{7} + 101736 x^{6} - 18128 x^{5} - 201043 x^{4} - 289242 x^{3} - 17626 x^{2} + 111368 x + 145993 \)

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:  \(1670627195488492909044967149666304=2^{30}\cdot 11^{18}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43} a^{18} - \frac{3}{43} a^{17} - \frac{17}{43} a^{16} - \frac{10}{43} a^{15} + \frac{18}{43} a^{14} - \frac{18}{43} a^{13} - \frac{16}{43} a^{12} + \frac{20}{43} a^{11} + \frac{9}{43} a^{10} - \frac{13}{43} a^{9} + \frac{21}{43} a^{8} + \frac{16}{43} a^{7} + \frac{12}{43} a^{6} - \frac{6}{43} a^{5} - \frac{12}{43} a^{4} - \frac{5}{43} a^{3} + \frac{3}{43} a^{2} - \frac{6}{43} a + \frac{6}{43}$, $\frac{1}{117539593275039432375337540688671064916534395191852097114761} a^{19} + \frac{20691457706853414082568721740451081491836865895373961542}{5110417098914757929362501769072654996371060660515308570207} a^{18} + \frac{20224018152850007385589482963803850572569487529185463019280}{117539593275039432375337540688671064916534395191852097114761} a^{17} + \frac{4150548123722069886005552907729230098512820728348992167689}{117539593275039432375337540688671064916534395191852097114761} a^{16} + \frac{54694714903189624606108372236257897953793185154773975027381}{117539593275039432375337540688671064916534395191852097114761} a^{15} + \frac{9615200222723418437687362379254244944466574670225576540989}{117539593275039432375337540688671064916534395191852097114761} a^{14} - \frac{29249669930826817652248406912144274729452699216544808590354}{117539593275039432375337540688671064916534395191852097114761} a^{13} - \frac{48299085648838604272558077004753731970868798651732268913283}{117539593275039432375337540688671064916534395191852097114761} a^{12} - \frac{325170122325863241720507881894665883761950885161034209553}{5110417098914757929362501769072654996371060660515308570207} a^{11} + \frac{55983398694996489901255132035833964973107867777259216034217}{117539593275039432375337540688671064916534395191852097114761} a^{10} - \frac{30838978958104451546205582263597178903720770905148722675513}{117539593275039432375337540688671064916534395191852097114761} a^{9} - \frac{66302013526406513411147165467471241355151063619485617988}{2733478913373010055240407922992350346896148725391909235227} a^{8} - \frac{471287834653289857107429756731008151869659601214171939606}{117539593275039432375337540688671064916534395191852097114761} a^{7} + \frac{14584005803970644366442736316969133782703659897804496035634}{117539593275039432375337540688671064916534395191852097114761} a^{6} + \frac{57831140185315957912285811656422705067809831389603791357856}{117539593275039432375337540688671064916534395191852097114761} a^{5} - \frac{52392242919781877220353622683691348000766999606210970233332}{117539593275039432375337540688671064916534395191852097114761} a^{4} - \frac{5143063789102055547003801665733627290904724416484305068967}{117539593275039432375337540688671064916534395191852097114761} a^{3} - \frac{37955248562799085174357396964272271581735354768428833439433}{117539593275039432375337540688671064916534395191852097114761} a^{2} - \frac{53575821297832441401818915563832402902529223541946168120255}{117539593275039432375337540688671064916534395191852097114761} a + \frac{34609433783817808012075584586789724706755209272469885136105}{117539593275039432375337540688671064916534395191852097114761}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{12}$, which has order $48$ (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:  \( 3932685.35305 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T74):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.0.55534384018432.1, 10.10.1277290832423936.1, 10.0.5048580365312.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
11Data not computed
$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$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$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}$