Properties

Label 19.1.990...639.1
Degree $19$
Signature $[1, 9]$
Discriminant $-9.905\times 10^{22}$
Root discriminant $16.23$
Ramified prime $359$
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 - 2*x^18 + 2*x^17 - 2*x^16 - 3*x^15 + 14*x^14 - 7*x^13 - 22*x^12 + 30*x^11 - 9*x^10 + 5*x^9 - 2*x^8 - 51*x^7 + 90*x^6 - 19*x^5 - 91*x^4 + 113*x^3 - 59*x^2 + 14*x - 1)
 
gp: K = bnfinit(x^19 - 2*x^18 + 2*x^17 - 2*x^16 - 3*x^15 + 14*x^14 - 7*x^13 - 22*x^12 + 30*x^11 - 9*x^10 + 5*x^9 - 2*x^8 - 51*x^7 + 90*x^6 - 19*x^5 - 91*x^4 + 113*x^3 - 59*x^2 + 14*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 14, -59, 113, -91, -19, 90, -51, -2, 5, -9, 30, -22, -7, 14, -3, -2, 2, -2, 1]);
 

\( x^{19} - 2 x^{18} + 2 x^{17} - 2 x^{16} - 3 x^{15} + 14 x^{14} - 7 x^{13} - 22 x^{12} + 30 x^{11} - 9 x^{10} + 5 x^{9} - 2 x^{8} - 51 x^{7} + 90 x^{6} - 19 x^{5} - 91 x^{4} + 113 x^{3} - 59 x^{2} + 14 x - 1 \)

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:  \(-99048986760825351881639\)\(\medspace = -\,359^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $16.23$
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:  $359$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{2}{7} a^{13} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{91} a^{16} - \frac{2}{91} a^{15} - \frac{2}{7} a^{14} + \frac{38}{91} a^{13} + \frac{6}{13} a^{12} + \frac{16}{91} a^{11} + \frac{3}{91} a^{10} + \frac{2}{91} a^{9} + \frac{4}{13} a^{8} + \frac{17}{91} a^{7} - \frac{18}{91} a^{6} + \frac{23}{91} a^{5} + \frac{25}{91} a^{4} + \frac{33}{91} a^{3} + \frac{31}{91} a^{2} - \frac{37}{91} a - \frac{9}{91}$, $\frac{1}{91} a^{17} - \frac{4}{91} a^{15} + \frac{12}{91} a^{14} - \frac{25}{91} a^{13} + \frac{22}{91} a^{12} - \frac{17}{91} a^{11} - \frac{5}{91} a^{10} - \frac{20}{91} a^{9} - \frac{31}{91} a^{8} - \frac{23}{91} a^{7} + \frac{3}{7} a^{6} + \frac{32}{91} a^{5} + \frac{18}{91} a^{4} + \frac{6}{91} a^{3} - \frac{27}{91} a^{2} + \frac{3}{13} a - \frac{31}{91}$, $\frac{1}{189007} a^{18} + \frac{108}{27001} a^{17} - \frac{202}{189007} a^{16} - \frac{9805}{189007} a^{15} + \frac{28391}{189007} a^{14} - \frac{74177}{189007} a^{13} + \frac{83374}{189007} a^{12} - \frac{16025}{189007} a^{11} - \frac{4202}{14539} a^{10} - \frac{1701}{27001} a^{9} - \frac{4226}{14539} a^{8} + \frac{94309}{189007} a^{7} - \frac{78941}{189007} a^{6} + \frac{401}{2077} a^{5} + \frac{83830}{189007} a^{4} + \frac{78237}{189007} a^{3} - \frac{77671}{189007} a^{2} + \frac{35274}{189007} a + \frac{27486}{189007}$

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:  \( 4771.45184867 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{9}\cdot 4771.45184867 \cdot 1}{2\sqrt{99048986760825351881639}}\approx 0.231389870009$

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 $19$ $19$ $19$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
359Data not computed