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Properties

Label 19.1.15807705321...7459.1
Degree $19$
Signature $[1, 9]$
Discriminant $-\,19^{33}$
Root discriminant $166.34$
Ramified prime $19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{19}:C_{6}$ (as 19T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2423978601, -656755330, 1883060322, 157805450, -1426640821, -1023131209, -342335673, -53817785, 7072465, 5016304, 941507, 196973, 45277, 8683, 570, -950, -171, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 171*x^16 - 950*x^15 + 570*x^14 + 8683*x^13 + 45277*x^12 + 196973*x^11 + 941507*x^10 + 5016304*x^9 + 7072465*x^8 - 53817785*x^7 - 342335673*x^6 - 1023131209*x^5 - 1426640821*x^4 + 157805450*x^3 + 1883060322*x^2 - 656755330*x - 2423978601)
 
gp: K = bnfinit(x^19 - 171*x^16 - 950*x^15 + 570*x^14 + 8683*x^13 + 45277*x^12 + 196973*x^11 + 941507*x^10 + 5016304*x^9 + 7072465*x^8 - 53817785*x^7 - 342335673*x^6 - 1023131209*x^5 - 1426640821*x^4 + 157805450*x^3 + 1883060322*x^2 - 656755330*x - 2423978601, 1)
 

Normalized defining polynomial

\( x^{19} - 171 x^{16} - 950 x^{15} + 570 x^{14} + 8683 x^{13} + 45277 x^{12} + 196973 x^{11} + 941507 x^{10} + 5016304 x^{9} + 7072465 x^{8} - 53817785 x^{7} - 342335673 x^{6} - 1023131209 x^{5} - 1426640821 x^{4} + 157805450 x^{3} + 1883060322 x^{2} - 656755330 x - 2423978601 \)

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

Invariants

Degree:  $19$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1580770532156861979997149793605296459437459=-\,19^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $166.34$
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:  $19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} - \frac{1}{11} a^{8} + \frac{3}{11} a^{7} - \frac{5}{11} a^{6} + \frac{1}{11} a^{4} - \frac{1}{11} a^{3} + \frac{3}{11} a^{2} - \frac{5}{11} a$, $\frac{1}{209} a^{10} - \frac{1}{209} a^{9} + \frac{47}{209} a^{8} + \frac{28}{209} a^{7} - \frac{2}{19} a^{6} - \frac{65}{209} a^{5} - \frac{12}{209} a^{4} - \frac{8}{209} a^{3} + \frac{6}{209} a^{2} - \frac{6}{19} a + \frac{5}{19}$, $\frac{1}{209} a^{11} + \frac{8}{209} a^{9} - \frac{96}{209} a^{8} + \frac{101}{209} a^{7} + \frac{103}{209} a^{6} - \frac{7}{19} a^{5} - \frac{58}{209} a^{4} + \frac{36}{209} a^{3} + \frac{35}{209} a^{2} - \frac{30}{209} a + \frac{5}{19}$, $\frac{1}{209} a^{12} + \frac{7}{209} a^{9} + \frac{48}{209} a^{8} - \frac{45}{209} a^{7} + \frac{42}{209} a^{6} + \frac{4}{19} a^{5} + \frac{18}{209} a^{4} + \frac{4}{209} a^{3} - \frac{2}{209} a^{2} - \frac{101}{209} a - \frac{2}{19}$, $\frac{1}{209} a^{13} - \frac{2}{209} a^{9} + \frac{101}{209} a^{8} + \frac{93}{209} a^{7} + \frac{65}{209} a^{6} + \frac{5}{19} a^{5} + \frac{31}{209} a^{4} - \frac{98}{209} a^{3} + \frac{104}{209} a^{2} + \frac{98}{209} a + \frac{3}{19}$, $\frac{1}{209} a^{14} + \frac{4}{209} a^{9} + \frac{73}{209} a^{8} + \frac{45}{209} a^{7} + \frac{68}{209} a^{6} - \frac{9}{19} a^{5} - \frac{8}{209} a^{4} - \frac{26}{209} a^{3} + \frac{34}{209} a^{2} - \frac{42}{209} a - \frac{9}{19}$, $\frac{1}{2299} a^{15} - \frac{1}{2299} a^{14} - \frac{2}{2299} a^{13} - \frac{1}{2299} a^{12} + \frac{3}{2299} a^{11} - \frac{2}{2299} a^{10} + \frac{96}{2299} a^{9} + \frac{197}{2299} a^{8} + \frac{435}{2299} a^{7} - \frac{943}{2299} a^{6} - \frac{740}{2299} a^{5} + \frac{9}{2299} a^{4} + \frac{408}{2299} a^{3} - \frac{1049}{2299} a^{2} - \frac{5}{209} a + \frac{6}{19}$, $\frac{1}{4598} a^{16} - \frac{1}{4598} a^{15} + \frac{9}{4598} a^{14} - \frac{1}{4598} a^{13} - \frac{4}{2299} a^{12} + \frac{9}{4598} a^{11} - \frac{3}{4598} a^{10} + \frac{71}{2299} a^{9} - \frac{96}{2299} a^{8} + \frac{29}{2299} a^{7} - \frac{348}{2299} a^{6} + \frac{1725}{4598} a^{5} - \frac{918}{2299} a^{4} - \frac{2281}{4598} a^{3} - \frac{45}{418} a^{2} - \frac{26}{209} a - \frac{9}{38}$, $\frac{1}{50578} a^{17} + \frac{5}{50578} a^{16} + \frac{3}{50578} a^{15} + \frac{97}{50578} a^{14} + \frac{37}{25289} a^{13} - \frac{105}{50578} a^{12} + \frac{117}{50578} a^{11} + \frac{7}{25289} a^{10} - \frac{4}{121} a^{9} - \frac{9776}{25289} a^{8} - \frac{4772}{25289} a^{7} + \frac{16051}{50578} a^{6} + \frac{998}{2299} a^{5} - \frac{21723}{50578} a^{4} + \frac{481}{2662} a^{3} + \frac{62}{2299} a^{2} - \frac{159}{418} a + \frac{7}{19}$, $\frac{1}{6411485532687507849804036682681311640660025987491193530762} a^{18} + \frac{2720688410621593436529523164645634406444053783265577}{6411485532687507849804036682681311640660025987491193530762} a^{17} - \frac{163155530841344519914963791532697123652051338230437114}{3205742766343753924902018341340655820330012993745596765381} a^{16} + \frac{386300080262664594862912242380332043142771850766900055}{3205742766343753924902018341340655820330012993745596765381} a^{15} + \frac{14522949632081196305970277803566232559077713436762541125}{6411485532687507849804036682681311640660025987491193530762} a^{14} + \frac{1719680023844102229107012964476020931312369109561907719}{3205742766343753924902018341340655820330012993745596765381} a^{13} - \frac{3419136521217834221737540313552995975731802622719994663}{6411485532687507849804036682681311640660025987491193530762} a^{12} + \frac{10087777528264943287662250188838053363554287125288636261}{6411485532687507849804036682681311640660025987491193530762} a^{11} + \frac{574763612697205629037298441348309395716013230560260513}{582862321153409804527639698425573785514547817044653957342} a^{10} - \frac{88565310376719041245973830523807601911006862653550820036}{3205742766343753924902018341340655820330012993745596765381} a^{9} + \frac{288434574013944548017979900515788765793473184123999859522}{3205742766343753924902018341340655820330012993745596765381} a^{8} + \frac{3003157875852955544199965077870715843115474248636005028191}{6411485532687507849804036682681311640660025987491193530762} a^{7} + \frac{78574067149185909594475646374060028169932682743247375937}{291431160576704902263819849212786892757273908522326978671} a^{6} - \frac{840013100981692479244863450636250581563022779522190989163}{3205742766343753924902018341340655820330012993745596765381} a^{5} + \frac{2814756015444140692585034726455477811493571196010204790011}{6411485532687507849804036682681311640660025987491193530762} a^{4} - \frac{11143682001563876403473980780623287695561141998612820837}{30676964271232094975138931496082830816555148265508103018} a^{3} + \frac{1263154410321195362488305993753092085118586422719409295}{26493741870609536569438168110253353887024900774756998061} a^{2} + \frac{1117814242737962736508613373802746787615054454796301921}{2408521988237230597221651646386668535184081888614272551} a + \frac{171596212421631634488844370275189304852165058440887843}{437913088770405563131209390252121551851651252475322282}$

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:  \( 413873076897000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{19}:C_3$ (as 19T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 114
The 9 conjugacy class representatives for $C_{19}:C_{6}$
Character table for $C_{19}:C_{6}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{19}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
19Data not computed