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Properties

Label 19.19.1978419655...0000.1
Degree $19$
Signature $[19, 0]$
Discriminant $2^{18}\cdot 5^{18}\cdot 19^{19}$
Root discriminant $168.31$
Ramified primes $2, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_{19}$ (as 19T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8731210, -37109375, 0, 111328125, 0, -97968750, 0, 39187500, 0, -8490625, 0, 1080625, 0, -83125, 0, 3800, 0, -95, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 95*x^17 + 3800*x^15 - 83125*x^13 + 1080625*x^11 - 8490625*x^9 + 39187500*x^7 - 97968750*x^5 + 111328125*x^3 - 37109375*x - 8731210)
 
gp: K = bnfinit(x^19 - 95*x^17 + 3800*x^15 - 83125*x^13 + 1080625*x^11 - 8490625*x^9 + 39187500*x^7 - 97968750*x^5 + 111328125*x^3 - 37109375*x - 8731210, 1)
 

Normalized defining polynomial

\( x^{19} - 95 x^{17} + 3800 x^{15} - 83125 x^{13} + 1080625 x^{11} - 8490625 x^{9} + 39187500 x^{7} - 97968750 x^{5} + 111328125 x^{3} - 37109375 x - 8731210 \)

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

Invariants

Degree:  $19$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[19, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1978419655660313589123979000000000000000000=2^{18}\cdot 5^{18}\cdot 19^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $168.31$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{24478} a^{10} - \frac{6923}{24478} a^{9} - \frac{25}{12239} a^{8} - \frac{6679}{24478} a^{7} + \frac{875}{24478} a^{6} + \frac{2273}{24478} a^{5} - \frac{3125}{12239} a^{4} - \frac{4954}{12239} a^{3} - \frac{8853}{24478} a^{2} + \frac{2623}{24478} a - \frac{3125}{12239}$, $\frac{1}{24478} a^{11} - \frac{55}{24478} a^{9} - \frac{10137}{24478} a^{8} + \frac{550}{12239} a^{7} - \frac{5323}{12239} a^{6} - \frac{9625}{24478} a^{5} - \frac{777}{12239} a^{4} + \frac{9897}{24478} a^{3} + \frac{3108}{12239} a^{2} - \frac{9897}{24478} a + \frac{4177}{12239}$, $\frac{1}{24478} a^{12} + \frac{373}{12239} a^{9} - \frac{825}{12239} a^{8} - \frac{10821}{24478} a^{7} - \frac{5228}{12239} a^{6} + \frac{1071}{24478} a^{5} + \frac{8839}{24478} a^{4} - \frac{104}{12239} a^{3} - \frac{3626}{12239} a^{2} + \frac{5751}{24478} a - \frac{529}{12239}$, $\frac{1}{24478} a^{13} - \frac{975}{12239} a^{9} + \frac{2001}{24478} a^{8} + \frac{1522}{12239} a^{7} + \frac{9227}{24478} a^{6} + \frac{2163}{24478} a^{5} + \frac{5736}{12239} a^{4} - \frac{4120}{12239} a^{3} + \frac{1029}{24478} a^{2} + \frac{212}{12239} a + \frac{5840}{12239}$, $\frac{1}{24478} a^{14} - \frac{10471}{24478} a^{9} + \frac{1728}{12239} a^{8} + \frac{7473}{24478} a^{7} - \frac{5047}{24478} a^{6} - \frac{5587}{12239} a^{5} - \frac{2848}{12239} a^{4} - \frac{6429}{24478} a^{3} - \frac{2968}{12239} a^{2} + \frac{5314}{12239} a + \frac{1272}{12239}$, $\frac{1}{24478} a^{15} - \frac{7919}{24478} a^{9} - \frac{2039}{24478} a^{8} - \frac{3605}{12239} a^{7} - \frac{3821}{24478} a^{6} + \frac{2271}{24478} a^{5} + \frac{3993}{24478} a^{4} + \frac{4819}{12239} a^{3} + \frac{9051}{24478} a^{2} + \frac{3661}{24478} a + \frac{5211}{12239}$, $\frac{1}{24478} a^{16} + \frac{2722}{12239} a^{9} - \frac{5756}{12239} a^{8} + \frac{1068}{12239} a^{7} + \frac{2061}{12239} a^{6} - \frac{5964}{12239} a^{5} + \frac{5202}{12239} a^{4} - \frac{411}{24478} a^{3} + \frac{873}{12239} a^{2} + \frac{137}{24478} a + \frac{383}{12239}$, $\frac{1}{24478} a^{17} + \frac{2829}{12239} a^{9} + \frac{2539}{12239} a^{8} - \frac{4855}{12239} a^{7} - \frac{1109}{12239} a^{6} - \frac{1209}{12239} a^{5} + \frac{169}{24478} a^{4} - \frac{4307}{12239} a^{3} - \frac{1313}{24478} a^{2} - \frac{4086}{12239} a + \frac{290}{12239}$, $\frac{1}{24478} a^{18} + \frac{5306}{12239} a^{9} + \frac{1966}{12239} a^{8} - \frac{3234}{12239} a^{7} - \frac{4306}{12239} a^{6} - \frac{9515}{24478} a^{5} + \frac{3827}{12239} a^{4} + \frac{3531}{24478} a^{3} + \frac{57}{12239} a^{2} - \frac{3343}{12239} a - \frac{4105}{12239}$

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:  $18$
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:  \( 5210002514550000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_{19}$ (as 19T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 342
The 19 conjugacy class representatives for $F_{19}$
Character table for $F_{19}$

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 R ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R $18{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
5Data not computed
19Data not computed