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Properties

Label 19.19.4379339948...6041.1
Degree $19$
Signature $[19, 0]$
Discriminant $11^{18}\cdot 31^{12}$
Root discriminant $84.82$
Ramified primes $11, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{19}:C_{3}$ (as 19T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1023, 6820, -56947, -118602, 661705, -391512, -833723, 801636, 349019, -479512, -60665, 134959, 4257, -20317, -77, 1650, 0, -66, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 66*x^17 + 1650*x^15 - 77*x^14 - 20317*x^13 + 4257*x^12 + 134959*x^11 - 60665*x^10 - 479512*x^9 + 349019*x^8 + 801636*x^7 - 833723*x^6 - 391512*x^5 + 661705*x^4 - 118602*x^3 - 56947*x^2 + 6820*x + 1023)
 
gp: K = bnfinit(x^19 - 66*x^17 + 1650*x^15 - 77*x^14 - 20317*x^13 + 4257*x^12 + 134959*x^11 - 60665*x^10 - 479512*x^9 + 349019*x^8 + 801636*x^7 - 833723*x^6 - 391512*x^5 + 661705*x^4 - 118602*x^3 - 56947*x^2 + 6820*x + 1023, 1)
 

Normalized defining polynomial

\( x^{19} - 66 x^{17} + 1650 x^{15} - 77 x^{14} - 20317 x^{13} + 4257 x^{12} + 134959 x^{11} - 60665 x^{10} - 479512 x^{9} + 349019 x^{8} + 801636 x^{7} - 833723 x^{6} - 391512 x^{5} + 661705 x^{4} - 118602 x^{3} - 56947 x^{2} + 6820 x + 1023 \)

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:  \(4379339948779445965938732832699226041=11^{18}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.82$
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:  $11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{96647827391144002298850207410179143} a^{18} + \frac{14128361692011533387163869958073798}{96647827391144002298850207410179143} a^{17} + \frac{13581269517942120863289065035858063}{96647827391144002298850207410179143} a^{16} - \frac{9760671827082630086081919175884827}{96647827391144002298850207410179143} a^{15} - \frac{2818293806323678681028359925839382}{96647827391144002298850207410179143} a^{14} - \frac{7667324676785815338178775153580403}{96647827391144002298850207410179143} a^{13} - \frac{1483281824965363516145916297708809}{96647827391144002298850207410179143} a^{12} - \frac{6668861322630717813304788934695902}{96647827391144002298850207410179143} a^{11} + \frac{48012873143070005692523074718738288}{96647827391144002298850207410179143} a^{10} + \frac{2281066022841056660201225004634894}{32215942463714667432950069136726381} a^{9} - \frac{30664145485667189132834895076226476}{96647827391144002298850207410179143} a^{8} - \frac{25453250433821156606862066104689238}{96647827391144002298850207410179143} a^{7} - \frac{272443604073308003012896352843621}{96647827391144002298850207410179143} a^{6} + \frac{16766857679077920328303636409956109}{96647827391144002298850207410179143} a^{5} + \frac{18695577020500217838603825378216038}{96647827391144002298850207410179143} a^{4} - \frac{9614844458820160843295758542142781}{32215942463714667432950069136726381} a^{3} - \frac{4833434946600792638549113901641307}{32215942463714667432950069136726381} a^{2} + \frac{41541232974804498208631151592175792}{96647827391144002298850207410179143} a + \frac{2734694671743825291817324056327211}{32215942463714667432950069136726381}$

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:  \( 4971598897600 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{19}:C_3$ (as 19T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 57
The 9 conjugacy class representatives for $C_{19}:C_{3}$
Character table for $C_{19}:C_{3}$

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 $19$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $19$ $19$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
11Data not computed
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$