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Properties

Label 20.4.82306730226...1609.3
Degree $20$
Signature $[4, 8]$
Discriminant $11^{18}\cdot 23^{6}$
Root discriminant $22.17$
Ramified primes $11, 23$
Class number $2$
Class group $[2]$
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![1, 11, 30, 44, -68, -176, 72, 264, 103, -506, -342, -176, -30, -143, -53, -11, 5, -11, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 - 11*x^17 + 5*x^16 - 11*x^15 - 53*x^14 - 143*x^13 - 30*x^12 - 176*x^11 - 342*x^10 - 506*x^9 + 103*x^8 + 264*x^7 + 72*x^6 - 176*x^5 - 68*x^4 + 44*x^3 + 30*x^2 + 11*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^18 - 11*x^17 + 5*x^16 - 11*x^15 - 53*x^14 - 143*x^13 - 30*x^12 - 176*x^11 - 342*x^10 - 506*x^9 + 103*x^8 + 264*x^7 + 72*x^6 - 176*x^5 - 68*x^4 + 44*x^3 + 30*x^2 + 11*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{18} - 11 x^{17} + 5 x^{16} - 11 x^{15} - 53 x^{14} - 143 x^{13} - 30 x^{12} - 176 x^{11} - 342 x^{10} - 506 x^{9} + 103 x^{8} + 264 x^{7} + 72 x^{6} - 176 x^{5} - 68 x^{4} + 44 x^{3} + 30 x^{2} + 11 x + 1 \)

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:  \(823067302269314181883621609=11^{18}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.17$
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:  $11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{82019336804540967132963113} a^{19} - \frac{7291156017893518311275451}{82019336804540967132963113} a^{18} - \frac{39913945032993475316477785}{82019336804540967132963113} a^{17} - \frac{33178280167671071789171680}{82019336804540967132963113} a^{16} - \frac{39268380742784598203266046}{82019336804540967132963113} a^{15} + \frac{18390643946807251905212307}{82019336804540967132963113} a^{14} - \frac{13066029614269049474743561}{82019336804540967132963113} a^{13} - \frac{36930254154391070542647017}{82019336804540967132963113} a^{12} - \frac{26624388401360197420763062}{82019336804540967132963113} a^{11} - \frac{84345504388785715554428}{752470979858174010394157} a^{10} + \frac{36545014700739480976106392}{82019336804540967132963113} a^{9} + \frac{15122119671716441334082765}{82019336804540967132963113} a^{8} - \frac{21666252456677849801251768}{82019336804540967132963113} a^{7} - \frac{1080425760423813008895147}{82019336804540967132963113} a^{6} + \frac{39613205058174397110717039}{82019336804540967132963113} a^{5} + \frac{21597766733866648242399923}{82019336804540967132963113} a^{4} - \frac{35062392300586185463637051}{82019336804540967132963113} a^{3} + \frac{7420529224469406778082030}{82019336804540967132963113} a^{2} + \frac{28485968003764457349535849}{82019336804540967132963113} a - \frac{11378141431233700952996220}{82019336804540967132963113}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 158801.256052 \)
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

\(\Q(\zeta_{11})^+\), 10.4.1247354328539.1, 10.6.54232796893.1, 10.4.2608104505127.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
11Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$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}$