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Properties

Label 20.14.1630238666...4375.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,3^{8}\cdot 5^{10}\cdot 239^{9}$
Root discriminant $40.80$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-829, 3297, 2245, -20061, 6277, 42381, -25253, -41649, 27710, 24276, -13643, -11359, 4384, 3858, -1092, -806, 200, 90, -22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 22*x^18 + 90*x^17 + 200*x^16 - 806*x^15 - 1092*x^14 + 3858*x^13 + 4384*x^12 - 11359*x^11 - 13643*x^10 + 24276*x^9 + 27710*x^8 - 41649*x^7 - 25253*x^6 + 42381*x^5 + 6277*x^4 - 20061*x^3 + 2245*x^2 + 3297*x - 829)
 
gp: K = bnfinit(x^20 - 4*x^19 - 22*x^18 + 90*x^17 + 200*x^16 - 806*x^15 - 1092*x^14 + 3858*x^13 + 4384*x^12 - 11359*x^11 - 13643*x^10 + 24276*x^9 + 27710*x^8 - 41649*x^7 - 25253*x^6 + 42381*x^5 + 6277*x^4 - 20061*x^3 + 2245*x^2 + 3297*x - 829, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 22 x^{18} + 90 x^{17} + 200 x^{16} - 806 x^{15} - 1092 x^{14} + 3858 x^{13} + 4384 x^{12} - 11359 x^{11} - 13643 x^{10} + 24276 x^{9} + 27710 x^{8} - 41649 x^{7} - 25253 x^{6} + 42381 x^{5} + 6277 x^{4} - 20061 x^{3} + 2245 x^{2} + 3297 x - 829 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-163023866649059818618082021484375=-\,3^{8}\cdot 5^{10}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.80$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{33} a^{18} - \frac{5}{33} a^{17} - \frac{7}{33} a^{16} + \frac{14}{33} a^{15} - \frac{16}{33} a^{14} + \frac{10}{33} a^{13} - \frac{8}{33} a^{12} + \frac{2}{11} a^{11} + \frac{8}{33} a^{10} + \frac{4}{11} a^{9} - \frac{4}{11} a^{8} - \frac{4}{11} a^{7} + \frac{14}{33} a^{6} - \frac{5}{33} a^{5} + \frac{5}{33} a^{4} - \frac{13}{33} a^{3} + \frac{4}{33} a^{2} + \frac{1}{33} a + \frac{7}{33}$, $\frac{1}{4195107182328348613503359535} a^{19} + \frac{33163774152702889896699173}{4195107182328348613503359535} a^{18} + \frac{125742369179188846124448563}{1398369060776116204501119845} a^{17} + \frac{1753477719625987564932164998}{4195107182328348613503359535} a^{16} - \frac{169681230054064180645178779}{381373380211668055773032685} a^{15} - \frac{202899031506131603638881168}{1398369060776116204501119845} a^{14} + \frac{187534944936493323145901728}{839021436465669722700671907} a^{13} - \frac{5119138349022233770353073}{20072283169035160830159615} a^{12} - \frac{383546077027183292178413306}{839021436465669722700671907} a^{11} - \frac{686487810834524674876194529}{4195107182328348613503359535} a^{10} - \frac{263097371770516090630959632}{1398369060776116204501119845} a^{9} + \frac{699150506534740126855609768}{1398369060776116204501119845} a^{8} - \frac{352766605127456561664816397}{4195107182328348613503359535} a^{7} - \frac{56146060622892021689763041}{127124460070556018591010895} a^{6} + \frac{323465384146237422136653072}{1398369060776116204501119845} a^{5} + \frac{926760201889044094064967253}{4195107182328348613503359535} a^{4} + \frac{399239526107061325372296851}{1398369060776116204501119845} a^{3} + \frac{254613364188121124053477756}{839021436465669722700671907} a^{2} - \frac{72547394601017936534657216}{839021436465669722700671907} a + \frac{217088905210912011676963132}{4195107182328348613503359535}$

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:  $16$
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:  \( 2645909749.65 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed