↑

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)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

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(y^19 - 2*y^18 + 2*y^17 - 2*y^16 - 3*y^15 + 14*y^14 - 7*y^13 - 22*y^12 + 30*y^11 - 9*y^10 + 5*y^9 - 2*y^8 - 51*y^7 + 90*y^6 - 19*y^5 - 91*y^4 + 113*y^3 - 59*y^2 + 14*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); 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);
 
oscar: Qx, x = PolynomialRing(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)
 

\( x^{19} - 2 x^{18} + 2 x^{17} - 2 x^{16} - 3 x^{15} + 14 x^{14} - 7 x^{13} - 22 x^{12} + 30 x^{11} + \cdots - 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $19$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-99048986760825351881639\) \(\medspace = -\,359^{9}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(16.23\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $359^{1/2}\approx 18.947295321496416$
Ramified primes:   \(359\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-359}) \)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

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

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{2907}{2077}a^{18}-\frac{272291}{189007}a^{17}+\frac{181281}{189007}a^{16}-\frac{262386}{189007}a^{15}-\frac{157769}{27001}a^{14}+\frac{2721171}{189007}a^{13}+\frac{1136732}{189007}a^{12}-\frac{5591938}{189007}a^{11}+\frac{2160644}{189007}a^{10}+\frac{1538116}{189007}a^{9}+\frac{2092385}{189007}a^{8}+\frac{1038536}{189007}a^{7}-\frac{1881606}{27001}a^{6}+\frac{10666164}{189007}a^{5}+\frac{739366}{14539}a^{4}-\frac{18172608}{189007}a^{3}+\frac{9161671}{189007}a^{2}-\frac{894716}{189007}a-\frac{243008}{189007}$, $\frac{323794}{189007}a^{18}-\frac{472204}{189007}a^{17}+\frac{351432}{189007}a^{16}-\frac{33372}{14539}a^{15}-\frac{1210669}{189007}a^{14}+\frac{3918363}{189007}a^{13}+\frac{68836}{189007}a^{12}-\frac{1060466}{27001}a^{11}+\frac{5322301}{189007}a^{10}+\frac{664116}{189007}a^{9}+\frac{2031808}{189007}a^{8}+\frac{241793}{189007}a^{7}-\frac{16756043}{189007}a^{6}+\frac{19546519}{189007}a^{5}+\frac{6368665}{189007}a^{4}-\frac{26711639}{189007}a^{3}+\frac{1554439}{14539}a^{2}-\frac{883712}{27001}a+\frac{608758}{189007}$, $\frac{237743}{189007}a^{18}-\frac{327653}{189007}a^{17}+\frac{216316}{189007}a^{16}-\frac{302255}{189007}a^{15}-\frac{71447}{14539}a^{14}+\frac{2819412}{189007}a^{13}+\frac{50915}{27001}a^{12}-\frac{5455062}{189007}a^{11}+\frac{482185}{27001}a^{10}+\frac{899050}{189007}a^{9}+\frac{1626546}{189007}a^{8}+\frac{386598}{189007}a^{7}-\frac{12399626}{189007}a^{6}+\frac{1018077}{14539}a^{5}+\frac{889389}{27001}a^{4}-\frac{2720905}{27001}a^{3}+\frac{1855084}{27001}a^{2}-\frac{3288434}{189007}a+\frac{63010}{189007}$, $\frac{35416}{189007}a^{18}+\frac{21941}{27001}a^{17}-\frac{86001}{189007}a^{16}+\frac{37136}{189007}a^{15}-\frac{320672}{189007}a^{14}-\frac{522049}{189007}a^{13}+\frac{1699289}{189007}a^{12}+\frac{802649}{189007}a^{11}-\frac{2984953}{189007}a^{10}+\frac{23616}{27001}a^{9}+\frac{1254052}{189007}a^{8}+\frac{2165231}{189007}a^{7}-\frac{4597}{14539}a^{6}-\frac{7296724}{189007}a^{5}+\frac{371086}{14539}a^{4}+\frac{6043042}{189007}a^{3}-\frac{7662773}{189007}a^{2}+\frac{2388959}{189007}a+\frac{103820}{189007}$, $\frac{137548}{189007}a^{18}-\frac{189801}{189007}a^{17}+\frac{123953}{189007}a^{16}-\frac{182506}{189007}a^{15}-\frac{536150}{189007}a^{14}+\frac{1621089}{189007}a^{13}+\frac{191006}{189007}a^{12}-\frac{453202}{27001}a^{11}+\frac{20245}{2077}a^{10}+\frac{443483}{189007}a^{9}+\frac{1086265}{189007}a^{8}+\frac{14437}{14539}a^{7}-\frac{1052554}{27001}a^{6}+\frac{7521224}{189007}a^{5}+\frac{275166}{14539}a^{4}-\frac{10789154}{189007}a^{3}+\frac{7293540}{189007}a^{2}-\frac{2307241}{189007}a+\frac{304936}{189007}$, $\frac{120097}{189007}a^{18}-\frac{17005}{27001}a^{17}+\frac{138925}{189007}a^{16}-\frac{9593}{14539}a^{15}-\frac{498371}{189007}a^{14}+\frac{1149188}{189007}a^{13}+\frac{236296}{189007}a^{12}-\frac{2168382}{189007}a^{11}+\frac{1603153}{189007}a^{10}-\frac{15788}{189007}a^{9}+\frac{459619}{189007}a^{8}+\frac{535980}{189007}a^{7}-\frac{5118424}{189007}a^{6}+\frac{5709685}{189007}a^{5}+\frac{2147109}{189007}a^{4}-\frac{8411003}{189007}a^{3}+\frac{927243}{27001}a^{2}-\frac{241164}{27001}a+\frac{45351}{189007}$, $\frac{48736}{189007}a^{18}-\frac{138646}{189007}a^{17}+\frac{27309}{189007}a^{16}-\frac{113248}{189007}a^{15}-\frac{13291}{14539}a^{14}+\frac{929342}{189007}a^{13}-\frac{30088}{27001}a^{12}-\frac{1863929}{189007}a^{11}+\frac{185174}{27001}a^{10}+\frac{521036}{189007}a^{9}+\frac{303497}{189007}a^{8}-\frac{558437}{189007}a^{7}-\frac{3705304}{189007}a^{6}+\frac{378361}{14539}a^{5}+\frac{214364}{27001}a^{4}-\frac{938839}{27001}a^{3}+\frac{586037}{27001}a^{2}-\frac{831343}{189007}a-\frac{125997}{189007}$, $\frac{244367}{189007}a^{18}-\frac{219552}{189007}a^{17}+\frac{153546}{189007}a^{16}-\frac{248783}{189007}a^{15}-\frac{154048}{27001}a^{14}+\frac{2330451}{189007}a^{13}+\frac{1279900}{189007}a^{12}-\frac{4774167}{189007}a^{11}+\frac{1591006}{189007}a^{10}+\frac{161672}{27001}a^{9}+\frac{1873490}{189007}a^{8}+\frac{1434855}{189007}a^{7}-\frac{11570567}{189007}a^{6}+\frac{8500796}{189007}a^{5}+\frac{8817635}{189007}a^{4}-\frac{15447661}{189007}a^{3}+\frac{7802362}{189007}a^{2}-\frac{633244}{189007}a-\frac{444257}{189007}$, $\frac{304936}{189007}a^{18}-\frac{472324}{189007}a^{17}+\frac{420071}{189007}a^{16}-\frac{69417}{27001}a^{15}-\frac{1097314}{189007}a^{14}+\frac{3732954}{189007}a^{13}-\frac{513463}{189007}a^{12}-\frac{6517586}{189007}a^{11}+\frac{5975666}{189007}a^{10}-\frac{902129}{189007}a^{9}+\frac{1968163}{189007}a^{8}+\frac{476393}{189007}a^{7}-\frac{2194865}{27001}a^{6}+\frac{20076362}{189007}a^{5}+\frac{132880}{14539}a^{4}-\frac{1859386}{14539}a^{3}+\frac{23668614}{189007}a^{2}-\frac{10697684}{189007}a+\frac{1961863}{189007}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4771.45184867 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 4771.45184867 \cdot 1}{2\cdot\sqrt{99048986760825351881639}}\cr\approx \mathstrut & 0.231389870009 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_{19}$ (as 19T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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 $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $19$ $19$ $19$ ${\href{/padicField/7.2.0.1}{2} }^{9}{,}\,{\href{/padicField/7.1.0.1}{1} }$ $19$ ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }$ $19$ ${\href{/padicField/19.2.0.1}{2} }^{9}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $19$ ${\href{/padicField/29.2.0.1}{2} }^{9}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{9}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $19$ $19$ ${\href{/padicField/43.2.0.1}{2} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }$ $19$ ${\href{/padicField/53.2.0.1}{2} }^{9}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{9}{,}\,{\href{/padicField/59.1.0.1}{1} }$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(359\) Copy content Toggle raw display $\Q_{359}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$