↑

Properties

Label 20.0.15709298240...8961.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{16}\cdot 43^{4}$
Root discriminant $14.45$
Ramified primes $11, 43$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:C_5$ (as 20T41)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 4, -7, 24, -18, 45, -54, 92, -77, 89, -77, 78, -62, 42, -29, 21, -10, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 5*x^18 - 10*x^17 + 21*x^16 - 29*x^15 + 42*x^14 - 62*x^13 + 78*x^12 - 77*x^11 + 89*x^10 - 77*x^9 + 92*x^8 - 54*x^7 + 45*x^6 - 18*x^5 + 24*x^4 - 7*x^3 + 4*x^2 + x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 5*x^18 - 10*x^17 + 21*x^16 - 29*x^15 + 42*x^14 - 62*x^13 + 78*x^12 - 77*x^11 + 89*x^10 - 77*x^9 + 92*x^8 - 54*x^7 + 45*x^6 - 18*x^5 + 24*x^4 - 7*x^3 + 4*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 5 x^{18} - 10 x^{17} + 21 x^{16} - 29 x^{15} + 42 x^{14} - 62 x^{13} + 78 x^{12} - 77 x^{11} + 89 x^{10} - 77 x^{9} + 92 x^{8} - 54 x^{7} + 45 x^{6} - 18 x^{5} + 24 x^{4} - 7 x^{3} + 4 x^{2} + x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(157092982407310367598961=11^{16}\cdot 43^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.45$
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:  $11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{2}{11} a^{14} + \frac{3}{11} a^{12} + \frac{3}{11} a^{11} + \frac{4}{11} a^{10} - \frac{5}{11} a^{9} + \frac{3}{11} a^{6} - \frac{4}{11} a^{5} - \frac{5}{11} a^{4} - \frac{1}{11} a^{3} + \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{4}{11} a^{14} + \frac{3}{11} a^{13} - \frac{3}{11} a^{12} - \frac{2}{11} a^{11} - \frac{2}{11} a^{10} - \frac{1}{11} a^{9} + \frac{3}{11} a^{7} + \frac{1}{11} a^{6} + \frac{3}{11} a^{5} - \frac{2}{11} a^{4} + \frac{2}{11} a^{3} + \frac{1}{11} a^{2} - \frac{3}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{17} - \frac{3}{11} a^{13} - \frac{1}{11} a^{12} - \frac{1}{11} a^{11} + \frac{4}{11} a^{10} + \frac{2}{11} a^{9} + \frac{3}{11} a^{8} + \frac{1}{11} a^{7} + \frac{4}{11} a^{6} + \frac{4}{11} a^{5} + \frac{4}{11} a^{4} - \frac{3}{11} a^{3} - \frac{3}{11} a^{2} - \frac{5}{11} a - \frac{4}{11}$, $\frac{1}{253} a^{18} - \frac{8}{253} a^{17} - \frac{10}{253} a^{16} + \frac{2}{253} a^{15} + \frac{41}{253} a^{14} + \frac{59}{253} a^{13} - \frac{89}{253} a^{12} - \frac{94}{253} a^{11} + \frac{108}{253} a^{10} - \frac{24}{253} a^{9} + \frac{76}{253} a^{8} - \frac{56}{253} a^{7} + \frac{56}{253} a^{6} + \frac{3}{23} a^{5} - \frac{47}{253} a^{4} - \frac{67}{253} a^{3} + \frac{75}{253} a^{2} - \frac{31}{253} a + \frac{65}{253}$, $\frac{1}{6985583} a^{19} + \frac{548}{635053} a^{18} + \frac{123213}{6985583} a^{17} + \frac{183258}{6985583} a^{16} - \frac{305402}{6985583} a^{15} + \frac{2564601}{6985583} a^{14} + \frac{1776630}{6985583} a^{13} - \frac{1259268}{6985583} a^{12} - \frac{2045233}{6985583} a^{11} + \frac{664468}{6985583} a^{10} - \frac{884077}{6985583} a^{9} - \frac{321894}{6985583} a^{8} - \frac{275659}{635053} a^{7} - \frac{1009544}{6985583} a^{6} - \frac{843380}{6985583} a^{5} - \frac{988157}{6985583} a^{4} - \frac{1126882}{6985583} a^{3} - \frac{2626801}{6985583} a^{2} - \frac{283766}{6985583} a - \frac{3230574}{6985583}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1838.7824 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T41):

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

\(\Q(\zeta_{11})^+\), 10.4.9217431883.1 x2, 10.2.396349570969.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$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}$
43Data not computed