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Properties

Label 19.1.101...431.1
Degree $19$
Signature $[1, 9]$
Discriminant $-1.016\times 10^{29}$
Root discriminant $33.63$
Ramified primes $3, 557$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{19}$ (as 19T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 5*x^18 + 11*x^17 - 10*x^16 + 3*x^15 + 3*x^14 - 80*x^13 + 481*x^12 - 1180*x^11 + 1220*x^10 + 888*x^9 - 5070*x^8 + 8854*x^7 - 9440*x^6 + 6893*x^5 - 3748*x^4 + 2592*x^3 - 2790*x^2 + 2673*x - 1053)
 
gp: K = bnfinit(x^19 - 5*x^18 + 11*x^17 - 10*x^16 + 3*x^15 + 3*x^14 - 80*x^13 + 481*x^12 - 1180*x^11 + 1220*x^10 + 888*x^9 - 5070*x^8 + 8854*x^7 - 9440*x^6 + 6893*x^5 - 3748*x^4 + 2592*x^3 - 2790*x^2 + 2673*x - 1053, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1053, 2673, -2790, 2592, -3748, 6893, -9440, 8854, -5070, 888, 1220, -1180, 481, -80, 3, 3, -10, 11, -5, 1]);
 

\(x^{19} - 5 x^{18} + 11 x^{17} - 10 x^{16} + 3 x^{15} + 3 x^{14} - 80 x^{13} + 481 x^{12} - 1180 x^{11} + 1220 x^{10} + 888 x^{9} - 5070 x^{8} + 8854 x^{7} - 9440 x^{6} + 6893 x^{5} - 3748 x^{4} + 2592 x^{3} - 2790 x^{2} + 2673 x - 1053\)  Toggle raw display

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

Invariants

Degree:  $19$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-101575284882268140616515967431\)\(\medspace = -\,3^{9}\cdot 557^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $33.63$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 557$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{9} a^{8} + \frac{1}{3} a^{4} - \frac{4}{9} a^{2}$, $\frac{1}{9} a^{9} + \frac{2}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{2}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{27} a^{11} - \frac{1}{27} a^{9} + \frac{1}{9} a^{7} + \frac{2}{27} a^{5} + \frac{1}{3} a^{4} + \frac{4}{27} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{12} - \frac{1}{27} a^{10} + \frac{2}{27} a^{6} - \frac{5}{27} a^{4} + \frac{1}{3} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{81} a^{13} + \frac{1}{81} a^{11} + \frac{1}{27} a^{10} - \frac{2}{81} a^{9} - \frac{1}{27} a^{8} + \frac{8}{81} a^{7} - \frac{1}{9} a^{6} - \frac{1}{81} a^{5} - \frac{4}{27} a^{4} + \frac{11}{81} a^{3} + \frac{7}{27} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{81} a^{14} + \frac{1}{81} a^{12} - \frac{2}{81} a^{10} - \frac{1}{81} a^{8} + \frac{1}{9} a^{7} - \frac{1}{81} a^{6} + \frac{1}{9} a^{5} + \frac{38}{81} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2}$, $\frac{1}{243} a^{15} + \frac{1}{243} a^{14} + \frac{1}{243} a^{13} + \frac{1}{243} a^{12} - \frac{2}{243} a^{11} - \frac{2}{243} a^{10} - \frac{10}{243} a^{9} + \frac{8}{243} a^{8} + \frac{35}{243} a^{7} + \frac{35}{243} a^{6} - \frac{34}{243} a^{5} - \frac{61}{243} a^{4} - \frac{8}{27} a^{3} + \frac{11}{27} a^{2} - \frac{1}{3}$, $\frac{1}{3159} a^{16} - \frac{2}{3159} a^{15} + \frac{16}{3159} a^{14} + \frac{10}{3159} a^{13} + \frac{1}{243} a^{12} + \frac{43}{3159} a^{11} + \frac{158}{3159} a^{10} - \frac{67}{3159} a^{9} - \frac{70}{3159} a^{8} - \frac{298}{3159} a^{7} + \frac{302}{3159} a^{6} + \frac{245}{3159} a^{5} + \frac{406}{1053} a^{4} + \frac{203}{1053} a^{3} + \frac{25}{351} a^{2} + \frac{10}{117} a - \frac{1}{3}$, $\frac{1}{161109} a^{17} + \frac{16}{161109} a^{16} - \frac{37}{53703} a^{15} - \frac{61}{53703} a^{14} - \frac{161}{53703} a^{13} - \frac{653}{53703} a^{12} - \frac{290}{161109} a^{11} - \frac{122}{161109} a^{10} + \frac{424}{53703} a^{9} + \frac{2878}{53703} a^{8} - \frac{331}{53703} a^{7} + \frac{34}{243} a^{6} - \frac{9530}{161109} a^{5} + \frac{70828}{161109} a^{4} + \frac{4753}{17901} a^{3} + \frac{3028}{17901} a^{2} - \frac{11}{117} a + \frac{52}{153}$, $\frac{1}{1302244047} a^{18} + \frac{428}{144693783} a^{17} - \frac{80209}{1302244047} a^{16} + \frac{793007}{434081349} a^{15} + \frac{1936471}{434081349} a^{14} - \frac{63280}{33390873} a^{13} + \frac{313963}{100172619} a^{12} - \frac{7269046}{434081349} a^{11} - \frac{67681507}{1302244047} a^{10} - \frac{23466806}{434081349} a^{9} + \frac{18566036}{434081349} a^{8} + \frac{1014310}{33390873} a^{7} - \frac{24995450}{1302244047} a^{6} - \frac{59695478}{434081349} a^{5} + \frac{630458807}{1302244047} a^{4} + \frac{2484602}{5359029} a^{3} - \frac{51382954}{144693783} a^{2} - \frac{6634339}{16077087} a + \frac{520313}{1236699}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 191261859.157 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{9}\cdot 191261859.157 \cdot 1}{2\sqrt{101575284882268140616515967431}}\approx 9.15910765549$ (assuming GRH)

Galois group

$D_{19}$ (as 19T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 38
The 11 conjugacy class representatives for $D_{19}$
Character table for $D_{19}$

Intermediate fields

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

Sibling fields

Galois closure: 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 $19$ R $19$ $19$ $19$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $19$ $19$ $19$ $19$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
557Data not computed