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Properties

Label 20.0.76299222860...0057.1
Degree $20$
Signature $[0, 10]$
Discriminant $19^{15}\cdot 43^{9}$
Root discriminant $49.45$
Ramified primes $19, 43$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_5:D_4$ (as 20T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20335, -53508, 56653, -114989, 175479, -111903, 81372, -56363, 21467, -3941, 516, -3325, 2040, 1894, -906, -529, 193, 60, -18, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 18*x^18 + 60*x^17 + 193*x^16 - 529*x^15 - 906*x^14 + 1894*x^13 + 2040*x^12 - 3325*x^11 + 516*x^10 - 3941*x^9 + 21467*x^8 - 56363*x^7 + 81372*x^6 - 111903*x^5 + 175479*x^4 - 114989*x^3 + 56653*x^2 - 53508*x + 20335)
 
gp: K = bnfinit(x^20 - 3*x^19 - 18*x^18 + 60*x^17 + 193*x^16 - 529*x^15 - 906*x^14 + 1894*x^13 + 2040*x^12 - 3325*x^11 + 516*x^10 - 3941*x^9 + 21467*x^8 - 56363*x^7 + 81372*x^6 - 111903*x^5 + 175479*x^4 - 114989*x^3 + 56653*x^2 - 53508*x + 20335, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 18 x^{18} + 60 x^{17} + 193 x^{16} - 529 x^{15} - 906 x^{14} + 1894 x^{13} + 2040 x^{12} - 3325 x^{11} + 516 x^{10} - 3941 x^{9} + 21467 x^{8} - 56363 x^{7} + 81372 x^{6} - 111903 x^{5} + 175479 x^{4} - 114989 x^{3} + 56653 x^{2} - 53508 x + 20335 \)

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:  \(7629922286089782470241764351830057=19^{15}\cdot 43^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.45$
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:  $19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{35} a^{17} - \frac{13}{35} a^{16} + \frac{9}{35} a^{15} - \frac{1}{5} a^{14} - \frac{6}{35} a^{13} - \frac{1}{5} a^{12} - \frac{1}{35} a^{11} - \frac{2}{5} a^{10} + \frac{3}{7} a^{9} - \frac{17}{35} a^{8} - \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{11}{35} a^{5} + \frac{17}{35} a^{4} + \frac{3}{7} a^{3} + \frac{2}{35} a^{2} - \frac{8}{35} a$, $\frac{1}{35} a^{18} + \frac{3}{7} a^{16} + \frac{1}{7} a^{15} + \frac{8}{35} a^{14} - \frac{3}{7} a^{13} + \frac{13}{35} a^{12} + \frac{8}{35} a^{11} + \frac{8}{35} a^{10} + \frac{3}{35} a^{9} - \frac{16}{35} a^{8} - \frac{6}{35} a^{6} + \frac{2}{5} a^{5} - \frac{9}{35} a^{4} - \frac{13}{35} a^{3} - \frac{17}{35} a^{2} + \frac{1}{35} a$, $\frac{1}{43920211455350891287193214445277439411449492753825} a^{19} + \frac{482728769583822835753627962560157486287420110504}{43920211455350891287193214445277439411449492753825} a^{18} - \frac{8587389755987694472046461720239305410484200232}{1756808458214035651487728577811097576457979710153} a^{17} - \frac{3186058465695874034208131127991242032825085416948}{8784042291070178257438642889055487882289898550765} a^{16} + \frac{10248845794605937465011772307162607090212677690288}{43920211455350891287193214445277439411449492753825} a^{15} - \frac{4310534583744739674043664042315835421623074110113}{43920211455350891287193214445277439411449492753825} a^{14} - \frac{162343422235561785045754444133724823709455056736}{6274315922192984469599030635039634201635641821975} a^{13} + \frac{2295735607741876864661200949427464690343496223771}{8784042291070178257438642889055487882289898550765} a^{12} - \frac{91075099764004001339482783667979612308467936678}{1254863184438596893919806127007926840327128364395} a^{11} + \frac{587506029397785803766426467509017205704570638872}{8784042291070178257438642889055487882289898550765} a^{10} + \frac{2414603814357382335285405166393745423044037797383}{6274315922192984469599030635039634201635641821975} a^{9} - \frac{20985030253818176774463718181208871623665995225654}{43920211455350891287193214445277439411449492753825} a^{8} - \frac{994810556674526212666275483545423208301364931568}{6274315922192984469599030635039634201635641821975} a^{7} + \frac{508597846183717340103843944493695978738751003526}{8784042291070178257438642889055487882289898550765} a^{6} + \frac{2688719524191699565378407914608957507764269040781}{6274315922192984469599030635039634201635641821975} a^{5} - \frac{15673824737483659414564937618317363371923605581299}{43920211455350891287193214445277439411449492753825} a^{4} + \frac{8486037273360576389392045998753774867566181160551}{43920211455350891287193214445277439411449492753825} a^{3} + \frac{16288681599515971897979062748274869989141862357573}{43920211455350891287193214445277439411449492753825} a^{2} - \frac{655611785871946054676114814736162148323744102066}{43920211455350891287193214445277439411449492753825} a - \frac{457337009948356990400583619277804519625944020388}{1254863184438596893919806127007926840327128364395}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 281681007.343 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_4$ (as 20T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.294937.1, 5.5.667489.1, 10.0.8465289737299.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$19$19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.8.6.2$x^{8} - 19 x^{4} + 5776$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
19.8.6.2$x^{8} - 19 x^{4} + 5776$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
43Data not computed