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Properties

Label 20.4.59699157574...1344.3
Degree $20$
Signature $[4, 8]$
Discriminant $2^{30}\cdot 11^{18}$
Root discriminant $24.48$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, -88, -286, -440, -242, 440, 1210, 66, -1661, 110, 1112, -446, -415, 532, -178, -32, 27, -8, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 4*x^18 - 8*x^17 + 27*x^16 - 32*x^15 - 178*x^14 + 532*x^13 - 415*x^12 - 446*x^11 + 1112*x^10 + 110*x^9 - 1661*x^8 + 66*x^7 + 1210*x^6 + 440*x^5 - 242*x^4 - 440*x^3 - 286*x^2 - 88*x - 11)
 
gp: K = bnfinit(x^20 - 2*x^19 + 4*x^18 - 8*x^17 + 27*x^16 - 32*x^15 - 178*x^14 + 532*x^13 - 415*x^12 - 446*x^11 + 1112*x^10 + 110*x^9 - 1661*x^8 + 66*x^7 + 1210*x^6 + 440*x^5 - 242*x^4 - 440*x^3 - 286*x^2 - 88*x - 11, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 4 x^{18} - 8 x^{17} + 27 x^{16} - 32 x^{15} - 178 x^{14} + 532 x^{13} - 415 x^{12} - 446 x^{11} + 1112 x^{10} + 110 x^{9} - 1661 x^{8} + 66 x^{7} + 1210 x^{6} + 440 x^{5} - 242 x^{4} - 440 x^{3} - 286 x^{2} - 88 x - 11 \)

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:  \(5969915757478328440239161344=2^{30}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.48$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{422215528132087933885080673499} a^{19} + \frac{172817341494888697070513626360}{422215528132087933885080673499} a^{18} + \frac{45073454168251086450382467105}{422215528132087933885080673499} a^{17} - \frac{183164120665641427394833753128}{422215528132087933885080673499} a^{16} + \frac{166179854398702566010596478616}{422215528132087933885080673499} a^{15} + \frac{54518922479062999656815164154}{422215528132087933885080673499} a^{14} + \frac{124692055237760134655587006958}{422215528132087933885080673499} a^{13} + \frac{139684011884153727362850373694}{422215528132087933885080673499} a^{12} - \frac{3291725832788331198434905535}{18357196875308171038481768413} a^{11} - \frac{89990633851584545086070107981}{422215528132087933885080673499} a^{10} - \frac{210536599398757513621512989372}{422215528132087933885080673499} a^{9} - \frac{157207573351912654499563267489}{422215528132087933885080673499} a^{8} - \frac{185514220628636580481291056212}{422215528132087933885080673499} a^{7} - \frac{64478718185369438244170339550}{422215528132087933885080673499} a^{6} + \frac{152149197910751895772721374925}{422215528132087933885080673499} a^{5} + \frac{15860340165504590006890244589}{422215528132087933885080673499} a^{4} + \frac{185841825562902944874561828284}{422215528132087933885080673499} a^{3} - \frac{18955726625089706525302013503}{422215528132087933885080673499} a^{2} - \frac{182840834072869333599565290281}{422215528132087933885080673499} a - \frac{96975800828472983143240503954}{422215528132087933885080673499}$

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:  \( 824130.084723 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T44):

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

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), 10.2.2414538435584.1, \(\Q(\zeta_{44})^+\), 10.2.219503494144.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$