Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - x^18 + 2*x^17 - 5*x^16 + 8*x^15 - 14*x^14 + 13*x^13 - 10*x^12 - x^11 + 9*x^10 - 18*x^9 + 25*x^8 - 10*x^7 - 4*x^6 + 38*x^5 - 42*x^4 + 37*x^3 - 16*x^2 + 4*x + 1)
gp: K = bnfinit(y^19 - y^18 + 2*y^17 - 5*y^16 + 8*y^15 - 14*y^14 + 13*y^13 - 10*y^12 - y^11 + 9*y^10 - 18*y^9 + 25*y^8 - 10*y^7 - 4*y^6 + 38*y^5 - 42*y^4 + 37*y^3 - 16*y^2 + 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - x^18 + 2*x^17 - 5*x^16 + 8*x^15 - 14*x^14 + 13*x^13 - 10*x^12 - x^11 + 9*x^10 - 18*x^9 + 25*x^8 - 10*x^7 - 4*x^6 + 38*x^5 - 42*x^4 + 37*x^3 - 16*x^2 + 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - x^18 + 2*x^17 - 5*x^16 + 8*x^15 - 14*x^14 + 13*x^13 - 10*x^12 - x^11 + 9*x^10 - 18*x^9 + 25*x^8 - 10*x^7 - 4*x^6 + 38*x^5 - 42*x^4 + 37*x^3 - 16*x^2 + 4*x + 1)
\( x^{19} - x^{18} + 2 x^{17} - 5 x^{16} + 8 x^{15} - 14 x^{14} + 13 x^{13} - 10 x^{12} - x^{11} + 9 x^{10} + \cdots + 1 \)
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: |
\(-27217203547650508966391\)
\(\medspace = -\,311^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.16\) | 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: | $311^{1/2}\approx 17.635192088548397$ | ||
| Ramified primes: |
\(311\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-311}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{391}a^{17}-\frac{2}{17}a^{16}-\frac{128}{391}a^{15}-\frac{179}{391}a^{14}-\frac{4}{23}a^{13}-\frac{19}{391}a^{12}-\frac{67}{391}a^{11}-\frac{160}{391}a^{10}+\frac{154}{391}a^{9}-\frac{174}{391}a^{8}+\frac{189}{391}a^{7}+\frac{133}{391}a^{6}+\frac{94}{391}a^{5}-\frac{65}{391}a^{4}-\frac{126}{391}a^{3}+\frac{48}{391}a^{2}-\frac{11}{23}a+\frac{158}{391}$, $\frac{1}{161454457}a^{18}+\frac{135026}{161454457}a^{17}+\frac{69148682}{161454457}a^{16}-\frac{655864}{161454457}a^{15}-\frac{133411}{161454457}a^{14}+\frac{38668402}{161454457}a^{13}-\frac{39486610}{161454457}a^{12}+\frac{1235520}{9497321}a^{11}+\frac{15740082}{161454457}a^{10}+\frac{11932252}{161454457}a^{9}-\frac{43903937}{161454457}a^{8}-\frac{39923095}{161454457}a^{7}-\frac{18000683}{161454457}a^{6}-\frac{2167161}{8497603}a^{5}+\frac{1272916}{8497603}a^{4}-\frac{16289027}{161454457}a^{3}-\frac{42424184}{161454457}a^{2}+\frac{23572120}{161454457}a-\frac{75616849}{161454457}$
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 $2$)
| 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: |
$\frac{9552374}{161454457}a^{18}-\frac{578572}{9497321}a^{17}+\frac{1277435}{9497321}a^{16}-\frac{68663940}{161454457}a^{15}+\frac{75622407}{161454457}a^{14}-\frac{152949097}{161454457}a^{13}+\frac{223309312}{161454457}a^{12}-\frac{149922205}{161454457}a^{11}+\frac{91119757}{161454457}a^{10}+\frac{63703383}{161454457}a^{9}-\frac{216398529}{161454457}a^{8}+\frac{300178867}{161454457}a^{7}-\frac{176981042}{161454457}a^{6}+\frac{5390268}{8497603}a^{5}+\frac{9631097}{8497603}a^{4}-\frac{724317584}{161454457}a^{3}+\frac{501394384}{161454457}a^{2}-\frac{470820495}{161454457}a+\frac{66379852}{161454457}$, $\frac{36699367}{161454457}a^{18}+\frac{28354794}{161454457}a^{17}+\frac{14672195}{161454457}a^{16}-\frac{116940704}{161454457}a^{15}+\frac{11286958}{161454457}a^{14}-\frac{67111522}{161454457}a^{13}-\frac{135579622}{161454457}a^{12}+\frac{102901400}{161454457}a^{11}-\frac{168153355}{161454457}a^{10}-\frac{62627721}{161454457}a^{9}-\frac{53669268}{161454457}a^{8}+\frac{92922838}{161454457}a^{7}+\frac{702898760}{161454457}a^{6}-\frac{6554085}{8497603}a^{5}+\frac{13020068}{8497603}a^{4}+\frac{29619067}{9497321}a^{3}-\frac{351166293}{161454457}a^{2}+\frac{29266664}{161454457}a+\frac{241351549}{161454457}$, $\frac{23972479}{161454457}a^{18}+\frac{72298934}{161454457}a^{17}-\frac{5392190}{161454457}a^{16}-\frac{27963616}{161454457}a^{15}-\frac{209459706}{161454457}a^{14}+\frac{189039615}{161454457}a^{13}-\frac{494371337}{161454457}a^{12}+\frac{246391261}{161454457}a^{11}-\frac{173710788}{161454457}a^{10}-\frac{270431273}{161454457}a^{9}+\frac{271384995}{161454457}a^{8}-\frac{378671239}{161454457}a^{7}+\frac{1193191926}{161454457}a^{6}+\frac{15122347}{8497603}a^{5}-\frac{851140}{369461}a^{4}+\frac{1746692711}{161454457}a^{3}-\frac{905194377}{161454457}a^{2}+\frac{298805194}{161454457}a+\frac{170666906}{161454457}$, $\frac{20288859}{161454457}a^{18}-\frac{22124451}{161454457}a^{17}+\frac{30021142}{161454457}a^{16}-\frac{4714836}{7019759}a^{15}+\frac{171115079}{161454457}a^{14}-\frac{246312415}{161454457}a^{13}+\frac{11741296}{7019759}a^{12}-\frac{7890337}{7019759}a^{11}+\frac{25594497}{161454457}a^{10}+\frac{184054547}{161454457}a^{9}-\frac{15297094}{7019759}a^{8}+\frac{515455259}{161454457}a^{7}-\frac{233584438}{161454457}a^{6}-\frac{549449}{499859}a^{5}+\frac{26095150}{8497603}a^{4}-\frac{871515821}{161454457}a^{3}+\frac{618054776}{161454457}a^{2}-\frac{542783761}{161454457}a+\frac{112009654}{161454457}$, $\frac{4296595}{161454457}a^{18}+\frac{66405892}{161454457}a^{17}-\frac{51719530}{161454457}a^{16}+\frac{37502566}{161454457}a^{15}-\frac{280952315}{161454457}a^{14}+\frac{414204001}{161454457}a^{13}-\frac{559022274}{161454457}a^{12}+\frac{445544336}{161454457}a^{11}-\frac{163404711}{161454457}a^{10}-\frac{18066814}{9497321}a^{9}+\frac{493295199}{161454457}a^{8}-\frac{654036692}{161454457}a^{7}+\frac{1121917471}{161454457}a^{6}+\frac{3453477}{8497603}a^{5}-\frac{57951033}{8497603}a^{4}+\frac{1735916336}{161454457}a^{3}-\frac{1488784343}{161454457}a^{2}+\frac{693493366}{161454457}a-\frac{42261904}{161454457}$, $\frac{38387420}{161454457}a^{18}-\frac{83252921}{161454457}a^{17}+\frac{30461582}{161454457}a^{16}-\frac{210189430}{161454457}a^{15}+\frac{431412683}{161454457}a^{14}-\frac{493419901}{161454457}a^{13}+\frac{542195299}{161454457}a^{12}-\frac{5034132}{9497321}a^{11}-\frac{258086221}{161454457}a^{10}+\frac{642530004}{161454457}a^{9}-\frac{704361013}{161454457}a^{8}+\frac{1026494660}{161454457}a^{7}-\frac{451123872}{161454457}a^{6}-\frac{63853029}{8497603}a^{5}+\frac{82269676}{8497603}a^{4}-\frac{2026164049}{161454457}a^{3}+\frac{520136554}{161454457}a^{2}+\frac{60798726}{161454457}a-\frac{160519193}{161454457}$, $\frac{45837945}{161454457}a^{18}-\frac{5497022}{161454457}a^{17}+\frac{31604629}{161454457}a^{16}-\frac{175587840}{161454457}a^{15}+\frac{165680094}{161454457}a^{14}-\frac{281667601}{161454457}a^{13}+\frac{80340321}{161454457}a^{12}+\frac{36252743}{161454457}a^{11}-\frac{201249008}{161454457}a^{10}+\frac{7887290}{7019759}a^{9}-\frac{315358679}{161454457}a^{8}+\frac{481336952}{161454457}a^{7}+\frac{505364281}{161454457}a^{6}-\frac{26123507}{8497603}a^{5}+\frac{50058708}{8497603}a^{4}-\frac{294267548}{161454457}a^{3}-\frac{24772548}{161454457}a^{2}-\frac{15928685}{161454457}a+\frac{102823245}{161454457}$, $\frac{44835}{412927}a^{18}+\frac{27114525}{161454457}a^{17}-\frac{1164944}{7019759}a^{16}-\frac{26726508}{161454457}a^{15}-\frac{83466150}{161454457}a^{14}+\frac{8809102}{9497321}a^{13}-\frac{338542898}{161454457}a^{12}+\frac{368567957}{161454457}a^{11}-\frac{253808579}{161454457}a^{10}+\frac{18820}{161454457}a^{9}+\frac{168870023}{161454457}a^{8}-\frac{310058493}{161454457}a^{7}+\frac{846958901}{161454457}a^{6}-\frac{18816063}{8497603}a^{5}-\frac{6247660}{8497603}a^{4}+\frac{923481388}{161454457}a^{3}-\frac{1140842072}{161454457}a^{2}+\frac{36521992}{9497321}a-\frac{209196504}{161454457}$, $\frac{29547729}{161454457}a^{18}-\frac{30723673}{161454457}a^{17}+\frac{22562648}{161454457}a^{16}-\frac{117245084}{161454457}a^{15}+\frac{205891228}{161454457}a^{14}-\frac{260788269}{161454457}a^{13}+\frac{141942206}{161454457}a^{12}+\frac{38175449}{161454457}a^{11}-\frac{268262132}{161454457}a^{10}+\frac{366535976}{161454457}a^{9}-\frac{337930130}{161454457}a^{8}+\frac{436042793}{161454457}a^{7}+\frac{112118631}{161454457}a^{6}-\frac{38639458}{8497603}a^{5}+\frac{57308945}{8497603}a^{4}-\frac{717364542}{161454457}a^{3}+\frac{41488657}{161454457}a^{2}+\frac{14290312}{7019759}a-\frac{319151643}{161454457}$
| 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: | \( 2166.17030371 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 2166.17030371 \cdot 1}{2\cdot\sqrt{27217203547650508966391}}\cr\approx \mathstrut & 0.200396283220 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^19 - x^18 + 2*x^17 - 5*x^16 + 8*x^15 - 14*x^14 + 13*x^13 - 10*x^12 - x^11 + 9*x^10 - 18*x^9 + 25*x^8 - 10*x^7 - 4*x^6 + 38*x^5 - 42*x^4 + 37*x^3 - 16*x^2 + 4*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 - x^18 + 2*x^17 - 5*x^16 + 8*x^15 - 14*x^14 + 13*x^13 - 10*x^12 - x^11 + 9*x^10 - 18*x^9 + 25*x^8 - 10*x^7 - 4*x^6 + 38*x^5 - 42*x^4 + 37*x^3 - 16*x^2 + 4*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 - x^18 + 2*x^17 - 5*x^16 + 8*x^15 - 14*x^14 + 13*x^13 - 10*x^12 - x^11 + 9*x^10 - 18*x^9 + 25*x^8 - 10*x^7 - 4*x^6 + 38*x^5 - 42*x^4 + 37*x^3 - 16*x^2 + 4*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 - x^18 + 2*x^17 - 5*x^16 + 8*x^15 - 14*x^14 + 13*x^13 - 10*x^12 - x^11 + 9*x^10 - 18*x^9 + 25*x^8 - 10*x^7 - 4*x^6 + 38*x^5 - 42*x^4 + 37*x^3 - 16*x^2 + 4*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
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 | $19$ | $19$ | $19$ | $19$ | ${\href{/padicField/11.2.0.1}{2} }^{9}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $19$ | ${\href{/padicField/17.2.0.1}{2} }^{9}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{9}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{9}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{9}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{9}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{9}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19$ | $19$ | ${\href{/padicField/59.2.0.1}{2} }^{9}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(311\)
| $\Q_{311}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |