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Properties

Label 19.19.1028752458...9168.1
Degree $19$
Signature $[19, 0]$
Discriminant $2^{27}\cdot 3^{18}\cdot 19^{19}$
Root discriminant $144.06$
Ramified primes $2, 3, 19$
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![-67446, -373977, 0, 1869885, 0, -2742498, 0, 1828332, 0, -660231, 0, 140049, 0, -17955, 0, 1368, 0, -57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 57*x^17 + 1368*x^15 - 17955*x^13 + 140049*x^11 - 660231*x^9 + 1828332*x^7 - 2742498*x^5 + 1869885*x^3 - 373977*x - 67446)
 
gp: K = bnfinit(x^19 - 57*x^17 + 1368*x^15 - 17955*x^13 + 140049*x^11 - 660231*x^9 + 1828332*x^7 - 2742498*x^5 + 1869885*x^3 - 373977*x - 67446, 1)
 

Normalized defining polynomial

\( x^{19} - 57 x^{17} + 1368 x^{15} - 17955 x^{13} + 140049 x^{11} - 660231 x^{9} + 1828332 x^{7} - 2742498 x^{5} + 1869885 x^{3} - 373977 x - 67446 \)

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:  \(102875245824411623227124507531682584199168=2^{27}\cdot 3^{18}\cdot 19^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.06$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{393} a^{10} + \frac{38}{393} a^{9} - \frac{10}{131} a^{8} + \frac{22}{393} a^{7} - \frac{26}{131} a^{6} + \frac{65}{131} a^{5} - \frac{57}{131} a^{4} - \frac{42}{131} a^{3} + \frac{20}{131} a^{2} + \frac{64}{131} a - \frac{31}{131}$, $\frac{1}{393} a^{11} - \frac{11}{131} a^{9} - \frac{17}{393} a^{8} + \frac{1}{131} a^{7} + \frac{5}{131} a^{6} - \frac{38}{131} a^{5} + \frac{28}{131} a^{4} + \frac{44}{131} a^{3} - \frac{41}{131} a^{2} + \frac{26}{131} a - \frac{1}{131}$, $\frac{1}{393} a^{12} + \frac{58}{393} a^{9} + \frac{61}{393} a^{8} - \frac{15}{131} a^{7} + \frac{21}{131} a^{6} - \frac{54}{131} a^{5} - \frac{3}{131} a^{4} + \frac{14}{131} a^{3} + \frac{31}{131} a^{2} + \frac{15}{131} a + \frac{25}{131}$, $\frac{1}{1179} a^{13} + \frac{28}{393} a^{9} + \frac{41}{393} a^{8} - \frac{55}{393} a^{7} + \frac{48}{131} a^{6} - \frac{35}{131} a^{5} + \frac{15}{131} a^{4} - \frac{51}{131} a^{3} + \frac{55}{131} a^{2} - \frac{50}{131} a - \frac{12}{131}$, $\frac{1}{1179} a^{14} + \frac{25}{393} a^{9} - \frac{1}{393} a^{8} + \frac{52}{393} a^{7} + \frac{38}{131} a^{6} + \frac{29}{131} a^{5} - \frac{27}{131} a^{4} + \frac{52}{131} a^{3} + \frac{45}{131} a^{2} + \frac{30}{131} a - \frac{49}{131}$, $\frac{1}{1179} a^{15} - \frac{34}{393} a^{9} + \frac{16}{393} a^{8} - \frac{43}{393} a^{7} + \frac{24}{131} a^{6} + \frac{51}{131} a^{5} + \frac{36}{131} a^{4} + \frac{47}{131} a^{3} + \frac{54}{131} a^{2} + \frac{54}{131} a - \frac{11}{131}$, $\frac{1}{1179} a^{16} - \frac{2}{393} a^{9} - \frac{5}{131} a^{8} + \frac{34}{393} a^{7} - \frac{47}{131} a^{6} + \frac{19}{131} a^{5} - \frac{57}{131} a^{4} - \frac{64}{131} a^{3} - \frac{52}{131} a^{2} - \frac{62}{131} a - \frac{6}{131}$, $\frac{1}{1179} a^{17} + \frac{61}{393} a^{9} - \frac{26}{393} a^{8} + \frac{34}{393} a^{7} - \frac{33}{131} a^{6} - \frac{58}{131} a^{5} - \frac{47}{131} a^{4} - \frac{5}{131} a^{3} - \frac{22}{131} a^{2} - \frac{9}{131} a - \frac{62}{131}$, $\frac{1}{1179} a^{18} + \frac{14}{393} a^{9} + \frac{10}{131} a^{8} - \frac{44}{131} a^{6} + \frac{49}{131} a^{5} - \frac{65}{131} a^{4} + \frac{51}{131} a^{3} - \frac{50}{131} a^{2} - \frac{36}{131} a + \frac{57}{131}$

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:  \( 1170215493390000 \) (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 R $18{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R $18{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $19$ $18{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
3Data not computed
19Data not computed