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Properties

Label 20.20.7303816416...9296.4
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $49.34$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7129, 14522, 61028, -156412, -83388, 405814, -34044, -477204, 146585, 302378, -121760, -110132, 47809, 23466, -9888, -2854, 1065, 178, -54, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 54*x^18 + 178*x^17 + 1065*x^16 - 2854*x^15 - 9888*x^14 + 23466*x^13 + 47809*x^12 - 110132*x^11 - 121760*x^10 + 302378*x^9 + 146585*x^8 - 477204*x^7 - 34044*x^6 + 405814*x^5 - 83388*x^4 - 156412*x^3 + 61028*x^2 + 14522*x - 7129)
 
gp: K = bnfinit(x^20 - 4*x^19 - 54*x^18 + 178*x^17 + 1065*x^16 - 2854*x^15 - 9888*x^14 + 23466*x^13 + 47809*x^12 - 110132*x^11 - 121760*x^10 + 302378*x^9 + 146585*x^8 - 477204*x^7 - 34044*x^6 + 405814*x^5 - 83388*x^4 - 156412*x^3 + 61028*x^2 + 14522*x - 7129, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 54 x^{18} + 178 x^{17} + 1065 x^{16} - 2854 x^{15} - 9888 x^{14} + 23466 x^{13} + 47809 x^{12} - 110132 x^{11} - 121760 x^{10} + 302378 x^{9} + 146585 x^{8} - 477204 x^{7} - 34044 x^{6} + 405814 x^{5} - 83388 x^{4} - 156412 x^{3} + 61028 x^{2} + 14522 x - 7129 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{12} - \frac{1}{11} a^{11} + \frac{1}{11} a^{10} - \frac{1}{11} a^{9} - \frac{4}{11} a^{8} - \frac{5}{11} a^{6} + \frac{5}{11} a^{5} - \frac{5}{11} a^{3} - \frac{1}{11} a^{2} - \frac{3}{11} a - \frac{5}{11}$, $\frac{1}{11} a^{14} - \frac{2}{11} a^{12} - \frac{5}{11} a^{9} - \frac{4}{11} a^{8} - \frac{5}{11} a^{7} + \frac{5}{11} a^{5} - \frac{5}{11} a^{4} + \frac{5}{11} a^{3} - \frac{4}{11} a^{2} + \frac{3}{11} a - \frac{5}{11}$, $\frac{1}{11} a^{15} - \frac{2}{11} a^{12} - \frac{2}{11} a^{11} - \frac{3}{11} a^{10} + \frac{5}{11} a^{9} - \frac{2}{11} a^{8} - \frac{5}{11} a^{6} + \frac{5}{11} a^{5} + \frac{5}{11} a^{4} - \frac{3}{11} a^{3} + \frac{1}{11} a^{2} + \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{4}{11} a^{12} - \frac{5}{11} a^{11} - \frac{4}{11} a^{10} - \frac{4}{11} a^{9} + \frac{3}{11} a^{8} - \frac{5}{11} a^{7} - \frac{5}{11} a^{6} + \frac{4}{11} a^{5} - \frac{3}{11} a^{4} + \frac{2}{11} a^{3} - \frac{2}{11} a^{2} - \frac{5}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{17} + \frac{2}{11} a^{12} + \frac{3}{11} a^{11} - \frac{1}{11} a^{9} + \frac{1}{11} a^{8} - \frac{5}{11} a^{7} - \frac{5}{11} a^{6} - \frac{5}{11} a^{5} + \frac{2}{11} a^{4} + \frac{2}{11} a^{2} + \frac{2}{11}$, $\frac{1}{11} a^{18} + \frac{5}{11} a^{12} + \frac{2}{11} a^{11} - \frac{3}{11} a^{10} + \frac{3}{11} a^{9} + \frac{3}{11} a^{8} - \frac{5}{11} a^{7} + \frac{5}{11} a^{6} + \frac{3}{11} a^{5} + \frac{1}{11} a^{3} + \frac{2}{11} a^{2} - \frac{3}{11} a - \frac{1}{11}$, $\frac{1}{456420660747691674356780127832657} a^{19} - \frac{20720573371232300007086570628778}{456420660747691674356780127832657} a^{18} - \frac{7223540829338648760772429158499}{456420660747691674356780127832657} a^{17} - \frac{12216085857369925462648160460772}{456420660747691674356780127832657} a^{16} - \frac{9750271120296882419218522096621}{456420660747691674356780127832657} a^{15} - \frac{14206182165955208671212370668331}{456420660747691674356780127832657} a^{14} + \frac{11177372797741376225329674586019}{456420660747691674356780127832657} a^{13} - \frac{123454771684895987493422201745153}{456420660747691674356780127832657} a^{12} - \frac{22730776185744742316877482454108}{456420660747691674356780127832657} a^{11} - \frac{159347513531055647588006161507435}{456420660747691674356780127832657} a^{10} - \frac{15544081812832565024815470050196}{456420660747691674356780127832657} a^{9} + \frac{8680573247882518214033302585558}{41492787340699243123343647984787} a^{8} + \frac{183316141985913490040996521815703}{456420660747691674356780127832657} a^{7} + \frac{13960771286414028399876975542651}{456420660747691674356780127832657} a^{6} - \frac{187428988565570325821276213019157}{456420660747691674356780127832657} a^{5} + \frac{952566445961695208274442916682}{456420660747691674356780127832657} a^{4} + \frac{136649207697783643312239118656168}{456420660747691674356780127832657} a^{3} + \frac{5843730608604725060734138375119}{41492787340699243123343647984787} a^{2} + \frac{7631169112889445483109166621170}{41492787340699243123343647984787} a - \frac{208393506198127899503660410127532}{456420660747691674356780127832657}$

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:  $19$
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:  \( 41333671085.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T40):

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

\(\Q(\zeta_{11})^+\), 10.10.5048580365312.1, 10.10.2670699013250048.1, 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
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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$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$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$