↑

Properties

Label 20.0.29521807615...8224.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 11^{16}\cdot 89^{4}$
Root discriminant $23.63$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23, 100, 380, 840, 582, -543, -881, 395, 1389, -330, -81, -352, 366, -153, 153, -29, 10, -26, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 6*x^18 - 26*x^17 + 10*x^16 - 29*x^15 + 153*x^14 - 153*x^13 + 366*x^12 - 352*x^11 - 81*x^10 - 330*x^9 + 1389*x^8 + 395*x^7 - 881*x^6 - 543*x^5 + 582*x^4 + 840*x^3 + 380*x^2 + 100*x + 23)
 
gp: K = bnfinit(x^20 - x^19 + 6*x^18 - 26*x^17 + 10*x^16 - 29*x^15 + 153*x^14 - 153*x^13 + 366*x^12 - 352*x^11 - 81*x^10 - 330*x^9 + 1389*x^8 + 395*x^7 - 881*x^6 - 543*x^5 + 582*x^4 + 840*x^3 + 380*x^2 + 100*x + 23, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 6 x^{18} - 26 x^{17} + 10 x^{16} - 29 x^{15} + 153 x^{14} - 153 x^{13} + 366 x^{12} - 352 x^{11} - 81 x^{10} - 330 x^{9} + 1389 x^{8} + 395 x^{7} - 881 x^{6} - 543 x^{5} + 582 x^{4} + 840 x^{3} + 380 x^{2} + 100 x + 23 \)

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:  \(2952180761584785045865268224=2^{10}\cdot 11^{16}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.63$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{10984685886323689887420086572352399} a^{19} + \frac{84426642254772115415007561228867}{255457811309853253195815966798893} a^{18} - \frac{1203727434632892515050885974801464}{10984685886323689887420086572352399} a^{17} - \frac{406966725836433589062751599758211}{10984685886323689887420086572352399} a^{16} - \frac{3818388231421020521907302920542043}{10984685886323689887420086572352399} a^{15} + \frac{240342613551547850451029613970133}{10984685886323689887420086572352399} a^{14} + \frac{699261316862667165833462373122613}{10984685886323689887420086572352399} a^{13} + \frac{2443055021796019765381477936740155}{10984685886323689887420086572352399} a^{12} + \frac{1597720349778323277995379783198433}{10984685886323689887420086572352399} a^{11} - \frac{1130804400679498483279016152349131}{10984685886323689887420086572352399} a^{10} + \frac{3762448791578458638720158911549939}{10984685886323689887420086572352399} a^{9} - \frac{89240214291302544470172293696466}{10984685886323689887420086572352399} a^{8} + \frac{128424085467006975499286714947268}{10984685886323689887420086572352399} a^{7} + \frac{1524724242462749595943985972240814}{10984685886323689887420086572352399} a^{6} - \frac{4296557796509088061651024650311249}{10984685886323689887420086572352399} a^{5} + \frac{4581184928877789506546384798507303}{10984685886323689887420086572352399} a^{4} + \frac{102893264295549296318375991039971}{10984685886323689887420086572352399} a^{3} - \frac{2124763910693039136654776439531165}{10984685886323689887420086572352399} a^{2} + \frac{736267098247912428255808649705756}{10984685886323689887420086572352399} a - \frac{10636435404105045648210918679517}{477595038535812603800873329232713}$

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:  $9$
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:  \( 234369.448447 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$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}$
89Data not computed