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Properties

Label 20.2.65861868411...0144.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 241^{3}$
Root discriminant $21.92$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43, 10, 279, -976, 1032, -680, 904, -1220, 911, -684, 631, -384, 217, -186, 98, -22, 19, -16, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 16*x^17 + 19*x^16 - 22*x^15 + 98*x^14 - 186*x^13 + 217*x^12 - 384*x^11 + 631*x^10 - 684*x^9 + 911*x^8 - 1220*x^7 + 904*x^6 - 680*x^5 + 1032*x^4 - 976*x^3 + 279*x^2 + 10*x + 43)
 
gp: K = bnfinit(x^20 - 16*x^17 + 19*x^16 - 22*x^15 + 98*x^14 - 186*x^13 + 217*x^12 - 384*x^11 + 631*x^10 - 684*x^9 + 911*x^8 - 1220*x^7 + 904*x^6 - 680*x^5 + 1032*x^4 - 976*x^3 + 279*x^2 + 10*x + 43, 1)
 

Normalized defining polynomial

\( x^{20} - 16 x^{17} + 19 x^{16} - 22 x^{15} + 98 x^{14} - 186 x^{13} + 217 x^{12} - 384 x^{11} + 631 x^{10} - 684 x^{9} + 911 x^{8} - 1220 x^{7} + 904 x^{6} - 680 x^{5} + 1032 x^{4} - 976 x^{3} + 279 x^{2} + 10 x + 43 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-658618684118710741619590144=-\,2^{10}\cdot 11^{16}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.92$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{46} a^{18} - \frac{3}{46} a^{17} + \frac{1}{46} a^{16} + \frac{5}{46} a^{15} - \frac{2}{23} a^{14} + \frac{19}{46} a^{13} - \frac{19}{46} a^{12} - \frac{5}{46} a^{11} - \frac{7}{46} a^{10} + \frac{11}{23} a^{9} - \frac{9}{46} a^{7} - \frac{5}{46} a^{6} - \frac{3}{23} a^{5} - \frac{2}{23} a^{4} + \frac{1}{46} a^{3} + \frac{3}{46} a^{2} + \frac{19}{46} a + \frac{7}{23}$, $\frac{1}{73132524238636910294479186} a^{19} - \frac{565361121696757375630259}{73132524238636910294479186} a^{18} + \frac{5095501634944553147393688}{36566262119318455147239593} a^{17} - \frac{17833893918008201022612415}{73132524238636910294479186} a^{16} + \frac{961918258792947226420002}{36566262119318455147239593} a^{15} - \frac{17988043442413003160315857}{36566262119318455147239593} a^{14} - \frac{10203804815980514747673998}{36566262119318455147239593} a^{13} + \frac{5932787470803245834604935}{36566262119318455147239593} a^{12} - \frac{6773792160663253706672948}{36566262119318455147239593} a^{11} - \frac{948881453044227826419439}{3179674966897256969325182} a^{10} - \frac{21858271023903333658825803}{73132524238636910294479186} a^{9} + \frac{32720921483061731334994007}{73132524238636910294479186} a^{8} - \frac{10711128335616524932962640}{36566262119318455147239593} a^{7} - \frac{15320131938703948929131818}{36566262119318455147239593} a^{6} + \frac{3880869333554594056814445}{36566262119318455147239593} a^{5} + \frac{13883597771281722905959343}{73132524238636910294479186} a^{4} + \frac{21178874667437006769163723}{73132524238636910294479186} a^{3} - \frac{2777546104153280521852551}{36566262119318455147239593} a^{2} + \frac{9977918179382451168845932}{36566262119318455147239593} a + \frac{9422816100096674793680320}{36566262119318455147239593}$

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:  $10$
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:  \( 149631.019211 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.51660490321.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.8$x^{10} + x^{8} - 2 x^{6} - 2 x^{4} + x^{2} + 33$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
241Data not computed