Properties

Label 19.1.467...879.1
Degree $19$
Signature $[1, 9]$
Discriminant $-4.676\times 10^{26}$
Root discriminant $25.33$
Ramified prime $919$
Class number $1$
Class group trivial
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 + 10*x^17 - 5*x^16 + 9*x^15 - 42*x^14 + 51*x^13 - 6*x^12 - 23*x^11 + 4*x^10 - 29*x^9 + 112*x^8 - 63*x^7 - 147*x^6 + 183*x^5 + 15*x^4 - 77*x^3 - 17*x^2 + 37*x + 1)
 
gp: K = bnfinit(x^19 - 5*x^18 + 10*x^17 - 5*x^16 + 9*x^15 - 42*x^14 + 51*x^13 - 6*x^12 - 23*x^11 + 4*x^10 - 29*x^9 + 112*x^8 - 63*x^7 - 147*x^6 + 183*x^5 + 15*x^4 - 77*x^3 - 17*x^2 + 37*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 37, -17, -77, 15, 183, -147, -63, 112, -29, 4, -23, -6, 51, -42, 9, -5, 10, -5, 1]);
 

\(x^{19} - 5 x^{18} + 10 x^{17} - 5 x^{16} + 9 x^{15} - 42 x^{14} + 51 x^{13} - 6 x^{12} - 23 x^{11} + 4 x^{10} - 29 x^{9} + 112 x^{8} - 63 x^{7} - 147 x^{6} + 183 x^{5} + 15 x^{4} - 77 x^{3} - 17 x^{2} + 37 x + 1\)  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:  \(-467562425055097089773569879\)\(\medspace = -\,919^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $25.33$
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:  $919$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{7} + \frac{2}{9} a^{6} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{63} a^{17} + \frac{1}{63} a^{16} - \frac{1}{63} a^{15} - \frac{1}{9} a^{14} + \frac{5}{63} a^{13} + \frac{2}{63} a^{12} - \frac{1}{63} a^{11} - \frac{4}{63} a^{10} - \frac{1}{7} a^{9} - \frac{1}{21} a^{8} - \frac{20}{63} a^{7} + \frac{22}{63} a^{6} + \frac{10}{63} a^{5} + \frac{22}{63} a^{4} - \frac{2}{63} a^{3} + \frac{4}{9} a^{2} - \frac{22}{63} a + \frac{26}{63}$, $\frac{1}{1681146971469} a^{18} - \frac{12482114905}{1681146971469} a^{17} - \frac{26042434111}{560382323823} a^{16} + \frac{1134363586}{88481419551} a^{15} - \frac{142234021442}{1681146971469} a^{14} + \frac{119912754793}{1681146971469} a^{13} + \frac{198152955184}{1681146971469} a^{12} + \frac{230017054447}{1681146971469} a^{11} + \frac{53305188299}{1681146971469} a^{10} - \frac{15535359472}{186794107941} a^{9} + \frac{11712170908}{240163853067} a^{8} + \frac{16274440718}{88481419551} a^{7} + \frac{814706226419}{1681146971469} a^{6} - \frac{36869152762}{88481419551} a^{5} - \frac{418133673766}{1681146971469} a^{4} - \frac{396229955110}{1681146971469} a^{3} - \frac{174508750715}{560382323823} a^{2} + \frac{66551965369}{240163853067} a - \frac{310401455248}{1681146971469}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$

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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1152808.1284 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{9}\cdot 1152808.1284 \cdot 1}{2\sqrt{467562425055097089773569879}}\approx 0.81368450161$

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 $19$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $19$ $19$ $19$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $19$

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
919Data not computed