Properties

Label 19.1.473...375.1
Degree $19$
Signature $[1, 9]$
Discriminant $-4.731\times 10^{28}$
Root discriminant $32.30$
Ramified primes $5, 307$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{19}$ (as 19T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 6*x^18 + 18*x^17 - 22*x^16 - 38*x^15 + 219*x^14 - 403*x^13 + 380*x^12 - 536*x^11 + 1378*x^10 - 1608*x^9 + 691*x^8 - 1073*x^7 + 1881*x^6 - 757*x^5 + 716*x^4 - 1464*x^3 + 513*x^2 - 10*x + 295)
 
gp: K = bnfinit(x^19 - 6*x^18 + 18*x^17 - 22*x^16 - 38*x^15 + 219*x^14 - 403*x^13 + 380*x^12 - 536*x^11 + 1378*x^10 - 1608*x^9 + 691*x^8 - 1073*x^7 + 1881*x^6 - 757*x^5 + 716*x^4 - 1464*x^3 + 513*x^2 - 10*x + 295, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![295, -10, 513, -1464, 716, -757, 1881, -1073, 691, -1608, 1378, -536, 380, -403, 219, -38, -22, 18, -6, 1]);
 

\( x^{19} - 6 x^{18} + 18 x^{17} - 22 x^{16} - 38 x^{15} + 219 x^{14} - 403 x^{13} + 380 x^{12} - 536 x^{11} + 1378 x^{10} - 1608 x^{9} + 691 x^{8} - 1073 x^{7} + 1881 x^{6} - 757 x^{5} + 716 x^{4} - 1464 x^{3} + 513 x^{2} - 10 x + 295 \)

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:  \(-47312447868976594992787109375\)\(\medspace = -\,5^{9}\cdot 307^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $32.30$
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:  $5, 307$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{1}{5} a^{5}$, $\frac{1}{25} a^{14} + \frac{1}{25} a^{13} - \frac{1}{25} a^{12} + \frac{1}{25} a^{11} + \frac{1}{25} a^{10} - \frac{7}{25} a^{9} - \frac{8}{25} a^{8} + \frac{3}{25} a^{7} + \frac{1}{5} a^{6} + \frac{6}{25} a^{5} - \frac{1}{25} a^{4} + \frac{11}{25} a^{3} + \frac{3}{25} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{25} a^{15} - \frac{2}{25} a^{13} + \frac{2}{25} a^{12} + \frac{2}{25} a^{10} - \frac{1}{25} a^{9} + \frac{11}{25} a^{8} + \frac{2}{25} a^{7} + \frac{6}{25} a^{6} - \frac{7}{25} a^{5} + \frac{12}{25} a^{4} - \frac{8}{25} a^{3} - \frac{3}{25} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{16} - \frac{1}{25} a^{13} - \frac{2}{25} a^{12} - \frac{1}{25} a^{11} + \frac{1}{25} a^{10} + \frac{7}{25} a^{9} + \frac{11}{25} a^{8} - \frac{3}{25} a^{7} + \frac{3}{25} a^{6} - \frac{6}{25} a^{5} - \frac{2}{5} a^{4} - \frac{11}{25} a^{3} + \frac{11}{25} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{2125} a^{17} - \frac{2}{425} a^{16} - \frac{1}{2125} a^{15} - \frac{23}{2125} a^{14} + \frac{198}{2125} a^{13} - \frac{151}{2125} a^{12} - \frac{91}{2125} a^{11} + \frac{48}{2125} a^{10} - \frac{22}{125} a^{9} + \frac{382}{2125} a^{8} - \frac{18}{85} a^{7} - \frac{752}{2125} a^{6} + \frac{172}{425} a^{5} - \frac{541}{2125} a^{4} + \frac{307}{2125} a^{3} - \frac{213}{2125} a^{2} - \frac{39}{425} a - \frac{84}{425}$, $\frac{1}{276668931032258445625} a^{18} + \frac{1077543578953981}{16274643001897555625} a^{17} + \frac{3530321274886689859}{276668931032258445625} a^{16} + \frac{84004114088477411}{11066757241290337825} a^{15} - \frac{4368765847203168238}{276668931032258445625} a^{14} + \frac{1563967919821888408}{55333786206451689125} a^{13} - \frac{23864274021734750333}{276668931032258445625} a^{12} - \frac{26137678161754615334}{276668931032258445625} a^{11} + \frac{17508476690343105892}{276668931032258445625} a^{10} + \frac{16891746580759516752}{39524133004608349375} a^{9} - \frac{84064749096527828396}{276668931032258445625} a^{8} - \frac{128668278579884908777}{276668931032258445625} a^{7} - \frac{23201001116058074789}{276668931032258445625} a^{6} + \frac{16072981228172406117}{39524133004608349375} a^{5} + \frac{18662799322598935704}{55333786206451689125} a^{4} + \frac{114846639972906045301}{276668931032258445625} a^{3} - \frac{27640445204467188881}{276668931032258445625} a^{2} + \frac{13055438820250404513}{55333786206451689125} a + \frac{8117422455413795502}{55333786206451689125}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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 (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 26027741.7153 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{9}\cdot 26027741.7153 \cdot 1}{2\sqrt{47312447868976594992787109375}}\approx 1.82628097294$ (assuming GRH)

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 $19$ $19$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $19$ $19$ $19$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
307Data not computed