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Properties

Label 19.1.714...419.1
Degree $19$
Signature $[1, 9]$
Discriminant $-7.142\times 10^{26}$
Root discriminant $25.90$
Ramified prime $19$
Class number $1$
Class group trivial
Galois group $C_{19}:C_{6}$ (as 19T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^19 + 19*x^15 - 19*x^14 - 19*x^13 + 171*x^12 + 19*x^11 - 266*x^10 + 19*x^9 + 475*x^8 + 741*x^7 + 437*x^6 + 38*x^5 - 38*x^4 - 19*x^3 + 1)
 
gp: K = bnfinit(x^19 + 19*x^15 - 19*x^14 - 19*x^13 + 171*x^12 + 19*x^11 - 266*x^10 + 19*x^9 + 475*x^8 + 741*x^7 + 437*x^6 + 38*x^5 - 38*x^4 - 19*x^3 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -19, -38, 38, 437, 741, 475, 19, -266, 19, 171, -19, -19, 19, 0, 0, 0, 1]);
 

\( x^{19} + 19 x^{15} - 19 x^{14} - 19 x^{13} + 171 x^{12} + 19 x^{11} - 266 x^{10} + 19 x^{9} + 475 x^{8} + 741 x^{7} + 437 x^{6} + 38 x^{5} - 38 x^{4} - 19 x^{3} + 1 \)

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:  \(-714209495693373205673756419\)\(\medspace = -\,19^{21}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $25.90$
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:  $19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{10} a^{16} + \frac{1}{5} a^{15} - \frac{1}{10} a^{14} + \frac{1}{10} a^{13} + \frac{1}{5} a^{12} - \frac{3}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{3}{10} a^{6} - \frac{1}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a + \frac{3}{10}$, $\frac{1}{370} a^{17} + \frac{6}{185} a^{16} - \frac{171}{370} a^{15} + \frac{51}{370} a^{14} + \frac{16}{185} a^{13} - \frac{123}{370} a^{12} - \frac{71}{185} a^{11} + \frac{69}{185} a^{10} - \frac{52}{185} a^{9} - \frac{14}{37} a^{8} + \frac{93}{370} a^{7} - \frac{41}{370} a^{6} + \frac{92}{185} a^{5} - \frac{91}{370} a^{4} - \frac{147}{370} a^{3} + \frac{15}{74} a^{2} - \frac{137}{370} a + \frac{15}{37}$, $\frac{1}{26521869359148268510} a^{18} - \frac{6826040825032609}{26521869359148268510} a^{17} + \frac{628721947265875046}{13260934679574134255} a^{16} + \frac{6413722722121603321}{13260934679574134255} a^{15} - \frac{1260785536523544687}{13260934679574134255} a^{14} - \frac{909665398604062088}{2652186935914826851} a^{13} - \frac{6616310951575836579}{26521869359148268510} a^{12} - \frac{396876292381548847}{5304373871829653702} a^{11} - \frac{1381901801247986266}{13260934679574134255} a^{10} + \frac{2427022988972449907}{13260934679574134255} a^{9} - \frac{4661541396092582077}{26521869359148268510} a^{8} - \frac{411399133396466912}{13260934679574134255} a^{7} - \frac{1276119457010125836}{2652186935914826851} a^{6} + \frac{608820512510773566}{2652186935914826851} a^{5} - \frac{3736187603432184098}{13260934679574134255} a^{4} + \frac{1977641008305862037}{26521869359148268510} a^{3} + \frac{480456212482391413}{26521869359148268510} a^{2} - \frac{5284893778945036104}{13260934679574134255} a + \frac{1201796264987315159}{5304373871829653702}$

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:  \( 1437945.22979 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{9}\cdot 1437945.22979 \cdot 1}{2\sqrt{714209495693373205673756419}}\approx 0.821198882384$

Galois group

$D_{19}:C_3$ (as 19T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 114
The 9 conjugacy class representatives for $C_{19}:C_{6}$
Character table for $C_{19}:C_{6}$

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
19Data not computed