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Properties

Label 19.1.173...823.1
Degree $19$
Signature $[1, 9]$
Discriminant $-1.733\times 10^{27}$
Root discriminant $27.14$
Ramified prime $1063$
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 + 9*x^17 - 10*x^16 + 8*x^15 - 45*x^14 + 39*x^13 + 7*x^12 + 42*x^11 - 36*x^10 - 60*x^9 + 9*x^8 + 55*x^7 + 63*x^6 + 15*x^5 - 88*x^4 - 7*x^3 + 27*x^2 + 51*x + 1)
 
gp: K = bnfinit(x^19 + 9*x^17 - 10*x^16 + 8*x^15 - 45*x^14 + 39*x^13 + 7*x^12 + 42*x^11 - 36*x^10 - 60*x^9 + 9*x^8 + 55*x^7 + 63*x^6 + 15*x^5 - 88*x^4 - 7*x^3 + 27*x^2 + 51*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 51, 27, -7, -88, 15, 63, 55, 9, -60, -36, 42, 7, 39, -45, 8, -10, 9, 0, 1]);
 

\( x^{19} + 9 x^{17} - 10 x^{16} + 8 x^{15} - 45 x^{14} + 39 x^{13} + 7 x^{12} + 42 x^{11} - 36 x^{10} - 60 x^{9} + 9 x^{8} + 55 x^{7} + 63 x^{6} + 15 x^{5} - 88 x^{4} - 7 x^{3} + 27 x^{2} + 51 x + 1 \)

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:  \(-1733003264116942402576542823\)\(\medspace = -\,1063^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $27.14$
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:  $1063$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{1}{9} a^{2} + \frac{1}{9}$, $\frac{1}{45} a^{15} - \frac{2}{45} a^{14} + \frac{2}{45} a^{13} - \frac{1}{45} a^{12} + \frac{7}{45} a^{11} + \frac{7}{45} a^{10} - \frac{1}{45} a^{9} - \frac{7}{45} a^{8} - \frac{19}{45} a^{7} + \frac{4}{9} a^{6} + \frac{16}{45} a^{5} + \frac{1}{45} a^{4} - \frac{7}{45} a^{3} + \frac{2}{45} a^{2} + \frac{2}{9} a - \frac{11}{45}$, $\frac{1}{45} a^{16} - \frac{2}{45} a^{14} - \frac{2}{45} a^{13} - \frac{1}{9} a^{12} + \frac{2}{15} a^{11} - \frac{2}{45} a^{10} + \frac{1}{45} a^{9} + \frac{2}{45} a^{8} - \frac{2}{5} a^{7} + \frac{11}{45} a^{6} - \frac{7}{45} a^{5} + \frac{1}{9} a^{4} + \frac{1}{15} a^{3} - \frac{16}{45} a^{2} - \frac{1}{45} a - \frac{4}{15}$, $\frac{1}{1845} a^{17} + \frac{2}{205} a^{16} - \frac{1}{205} a^{15} - \frac{14}{1845} a^{14} + \frac{1}{369} a^{13} - \frac{29}{615} a^{12} + \frac{23}{205} a^{11} + \frac{31}{1845} a^{10} + \frac{18}{205} a^{9} + \frac{3}{205} a^{8} + \frac{11}{41} a^{7} - \frac{274}{1845} a^{6} + \frac{247}{1845} a^{5} + \frac{62}{615} a^{4} - \frac{47}{205} a^{3} + \frac{887}{1845} a^{2} - \frac{101}{369} a + \frac{14}{205}$, $\frac{1}{5334406065} a^{18} + \frac{468139}{5334406065} a^{17} + \frac{4847900}{1066881213} a^{16} + \frac{1060909}{197570595} a^{15} - \frac{141032008}{5334406065} a^{14} + \frac{117424238}{5334406065} a^{13} - \frac{33017746}{5334406065} a^{12} - \frac{61097681}{592711785} a^{11} + \frac{751016}{57359205} a^{10} - \frac{228245572}{1778135355} a^{9} + \frac{29836781}{197570595} a^{8} - \frac{182711074}{592711785} a^{7} - \frac{1444961006}{5334406065} a^{6} - \frac{2211984968}{5334406065} a^{5} + \frac{2559777664}{5334406065} a^{4} - \frac{28684102}{118542357} a^{3} + \frac{209721638}{5334406065} a^{2} - \frac{148281941}{1066881213} a - \frac{1803923422}{5334406065}$

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 $2$)
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:  \( 2724154.30052 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{9}\cdot 2724154.30052 \cdot 1}{2\sqrt{1733003264116942402576542823}}\approx 0.998736283786$

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 $19$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $19$ $19$ $19$ $19$ $19$ $19$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1063Data not computed