Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^19 + 9*x^17 - 10*x^16 + 8*x^15 - 45*x^14 + 39*x^13 + 7*x^12 + 42*x^11 - 36*x^10 - 60*x^9 + 9*x^8 + 55*x^7 + 63*x^6 + 15*x^5 - 88*x^4 - 7*x^3 + 27*x^2 + 51*x + 1)
gp: K = bnfinit(y^19 + 9*y^17 - 10*y^16 + 8*y^15 - 45*y^14 + 39*y^13 + 7*y^12 + 42*y^11 - 36*y^10 - 60*y^9 + 9*y^8 + 55*y^7 + 63*y^6 + 15*y^5 - 88*y^4 - 7*y^3 + 27*y^2 + 51*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 + 9*x^17 - 10*x^16 + 8*x^15 - 45*x^14 + 39*x^13 + 7*x^12 + 42*x^11 - 36*x^10 - 60*x^9 + 9*x^8 + 55*x^7 + 63*x^6 + 15*x^5 - 88*x^4 - 7*x^3 + 27*x^2 + 51*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 + 9*x^17 - 10*x^16 + 8*x^15 - 45*x^14 + 39*x^13 + 7*x^12 + 42*x^11 - 36*x^10 - 60*x^9 + 9*x^8 + 55*x^7 + 63*x^6 + 15*x^5 - 88*x^4 - 7*x^3 + 27*x^2 + 51*x + 1)
\( x^{19} + 9 x^{17} - 10 x^{16} + 8 x^{15} - 45 x^{14} + 39 x^{13} + 7 x^{12} + 42 x^{11} - 36 x^{10} + \cdots + 1 \)
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: |
\(-1733003264116942402576542823\)
\(\medspace = -\,1063^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.14\) | 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: | $1063^{1/2}\approx 32.60368077380221$ | ||
| Ramified primes: |
\(1063\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1063}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{12}+\frac{1}{9}a^{9}-\frac{1}{9}a^{8}-\frac{1}{9}a^{5}+\frac{1}{9}a^{4}-\frac{1}{9}a+\frac{1}{9}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{12}+\frac{1}{9}a^{10}-\frac{1}{9}a^{8}-\frac{1}{9}a^{6}+\frac{1}{9}a^{4}-\frac{1}{9}a^{2}+\frac{1}{9}$, $\frac{1}{45}a^{15}-\frac{2}{45}a^{14}+\frac{2}{45}a^{13}-\frac{1}{45}a^{12}+\frac{7}{45}a^{11}+\frac{7}{45}a^{10}-\frac{1}{45}a^{9}-\frac{7}{45}a^{8}-\frac{19}{45}a^{7}+\frac{4}{9}a^{6}+\frac{16}{45}a^{5}+\frac{1}{45}a^{4}-\frac{7}{45}a^{3}+\frac{2}{45}a^{2}+\frac{2}{9}a-\frac{11}{45}$, $\frac{1}{45}a^{16}-\frac{2}{45}a^{14}-\frac{2}{45}a^{13}-\frac{1}{9}a^{12}+\frac{2}{15}a^{11}-\frac{2}{45}a^{10}+\frac{1}{45}a^{9}+\frac{2}{45}a^{8}-\frac{2}{5}a^{7}+\frac{11}{45}a^{6}-\frac{7}{45}a^{5}+\frac{1}{9}a^{4}+\frac{1}{15}a^{3}-\frac{16}{45}a^{2}-\frac{1}{45}a-\frac{4}{15}$, $\frac{1}{1845}a^{17}+\frac{2}{205}a^{16}-\frac{1}{205}a^{15}-\frac{14}{1845}a^{14}+\frac{1}{369}a^{13}-\frac{29}{615}a^{12}+\frac{23}{205}a^{11}+\frac{31}{1845}a^{10}+\frac{18}{205}a^{9}+\frac{3}{205}a^{8}+\frac{11}{41}a^{7}-\frac{274}{1845}a^{6}+\frac{247}{1845}a^{5}+\frac{62}{615}a^{4}-\frac{47}{205}a^{3}+\frac{887}{1845}a^{2}-\frac{101}{369}a+\frac{14}{205}$, $\frac{1}{5334406065}a^{18}+\frac{468139}{5334406065}a^{17}+\frac{4847900}{1066881213}a^{16}+\frac{1060909}{197570595}a^{15}-\frac{141032008}{5334406065}a^{14}+\frac{117424238}{5334406065}a^{13}-\frac{33017746}{5334406065}a^{12}-\frac{61097681}{592711785}a^{11}+\frac{751016}{57359205}a^{10}-\frac{228245572}{1778135355}a^{9}+\frac{29836781}{197570595}a^{8}-\frac{182711074}{592711785}a^{7}-\frac{1444961006}{5334406065}a^{6}-\frac{2211984968}{5334406065}a^{5}+\frac{2559777664}{5334406065}a^{4}-\frac{28684102}{118542357}a^{3}+\frac{209721638}{5334406065}a^{2}-\frac{148281941}{1066881213}a-\frac{1803923422}{5334406065}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{7852838}{355627071}a^{18}+\frac{51837214}{1778135355}a^{17}+\frac{336735067}{1778135355}a^{16}+\frac{23958509}{592711785}a^{15}-\frac{321875347}{1778135355}a^{14}-\frac{1234807609}{1778135355}a^{13}-\frac{691728046}{1778135355}a^{12}+\frac{825885683}{592711785}a^{11}+\frac{326298}{424883}a^{10}-\frac{7450238}{118542357}a^{9}-\frac{1231684151}{592711785}a^{8}-\frac{35807174}{592711785}a^{7}+\frac{1139475199}{1778135355}a^{6}+\frac{4988462701}{1778135355}a^{5}+\frac{1458274372}{1778135355}a^{4}-\frac{1024883921}{592711785}a^{3}-\frac{1848422086}{1778135355}a^{2}+\frac{138227758}{355627071}a+\frac{1379399681}{1778135355}$, $\frac{1880524}{5334406065}a^{18}-\frac{131068697}{5334406065}a^{17}-\frac{90561332}{5334406065}a^{16}-\frac{25977422}{118542357}a^{15}+\frac{444593654}{5334406065}a^{14}+\frac{65877085}{1066881213}a^{13}+\frac{5697128876}{5334406065}a^{12}+\frac{3858722}{65856865}a^{11}-\frac{64372486}{57359205}a^{10}-\frac{2830324018}{1778135355}a^{9}-\frac{91725788}{197570595}a^{8}+\frac{155588951}{65856865}a^{7}+\frac{1756665011}{1066881213}a^{6}-\frac{1050175301}{5334406065}a^{5}-\frac{8996860391}{5334406065}a^{4}-\frac{1170237883}{592711785}a^{3}+\frac{1492027931}{5334406065}a^{2}+\frac{4890132008}{5334406065}a+\frac{4201872734}{5334406065}$, $\frac{40474538}{1778135355}a^{18}+\frac{6379151}{1778135355}a^{17}+\frac{72410884}{355627071}a^{16}-\frac{36026746}{197570595}a^{15}+\frac{276656713}{1778135355}a^{14}-\frac{1507367741}{1778135355}a^{13}+\frac{1441004011}{1778135355}a^{12}+\frac{18296197}{39514119}a^{11}+\frac{4574137}{6373245}a^{10}-\frac{103681435}{118542357}a^{9}-\frac{1862863}{1606265}a^{8}+\frac{74427779}{197570595}a^{7}+\frac{3188190944}{1778135355}a^{6}+\frac{2676149231}{1778135355}a^{5}-\frac{140707411}{1778135355}a^{4}-\frac{300071491}{197570595}a^{3}-\frac{29994955}{355627071}a^{2}+\frac{3011765482}{1778135355}a+\frac{2199336229}{1778135355}$, $\frac{255645259}{5334406065}a^{18}+\frac{115834726}{5334406065}a^{17}+\frac{2346800962}{5334406065}a^{16}-\frac{16009537}{65856865}a^{15}+\frac{1362691307}{5334406065}a^{14}-\frac{1834192685}{1066881213}a^{13}+\frac{686679694}{1066881213}a^{12}+\frac{473879842}{592711785}a^{11}+\frac{8516341}{11471841}a^{10}-\frac{326162566}{1778135355}a^{9}-\frac{73221340}{39514119}a^{8}+\frac{397487459}{592711785}a^{7}+\frac{15598232278}{5334406065}a^{6}+\frac{1272157562}{1066881213}a^{5}+\frac{2953622782}{5334406065}a^{4}-\frac{1582248014}{592711785}a^{3}-\frac{1872087178}{5334406065}a^{2}+\frac{17051730923}{5334406065}a+\frac{5720637947}{5334406065}$, $\frac{108698507}{5334406065}a^{18}-\frac{2758463}{1066881213}a^{17}+\frac{217867045}{1066881213}a^{16}-\frac{12958246}{65856865}a^{15}+\frac{2137365631}{5334406065}a^{14}-\frac{900047899}{1066881213}a^{13}+\frac{5438683768}{5334406065}a^{12}-\frac{358469557}{592711785}a^{11}+\frac{25481644}{57359205}a^{10}-\frac{3240230}{8673831}a^{9}-\frac{25975573}{197570595}a^{8}+\frac{702947953}{592711785}a^{7}+\frac{266321249}{5334406065}a^{6}-\frac{1949649529}{5334406065}a^{5}+\frac{1642066511}{5334406065}a^{4}-\frac{331760509}{592711785}a^{3}+\frac{9784506868}{5334406065}a^{2}-\frac{489796694}{5334406065}a+\frac{2581971964}{5334406065}$, $\frac{59970374}{5334406065}a^{18}+\frac{127874831}{5334406065}a^{17}+\frac{573559679}{5334406065}a^{16}+\frac{29057516}{197570595}a^{15}-\frac{89694871}{1066881213}a^{14}+\frac{279188104}{5334406065}a^{13}-\frac{5049564719}{5334406065}a^{12}+\frac{784281722}{592711785}a^{11}-\frac{78536702}{57359205}a^{10}+\frac{808908263}{355627071}a^{9}-\frac{36146761}{14456385}a^{8}+\frac{1047888391}{592711785}a^{7}-\frac{1057107881}{1066881213}a^{6}+\frac{1784053783}{1066881213}a^{5}+\frac{2295164831}{5334406065}a^{4}+\frac{169952993}{592711785}a^{3}+\frac{4450886683}{5334406065}a^{2}+\frac{7932900106}{5334406065}a+\frac{5745680809}{5334406065}$, $\frac{184666958}{5334406065}a^{18}-\frac{13329778}{5334406065}a^{17}+\frac{1612122608}{5334406065}a^{16}-\frac{216257263}{592711785}a^{15}+\frac{1215411877}{5334406065}a^{14}-\frac{7807548434}{5334406065}a^{13}+\frac{7733551288}{5334406065}a^{12}+\frac{226000456}{592711785}a^{11}+\frac{66867442}{57359205}a^{10}-\frac{3277064273}{1778135355}a^{9}-\frac{332961758}{197570595}a^{8}+\frac{318762638}{592711785}a^{7}+\frac{2441741740}{1066881213}a^{6}+\frac{14298152099}{5334406065}a^{5}-\frac{5902067164}{5334406065}a^{4}-\frac{458759827}{197570595}a^{3}-\frac{6565425626}{5334406065}a^{2}+\frac{14060218177}{5334406065}a+\frac{5811772939}{5334406065}$, $\frac{1749629}{172077615}a^{18}-\frac{54866}{4197015}a^{17}+\frac{13073531}{172077615}a^{16}-\frac{3950332}{19119735}a^{15}+\frac{12298633}{172077615}a^{14}-\frac{49906898}{172077615}a^{13}+\frac{127170388}{172077615}a^{12}+\frac{7610974}{19119735}a^{11}-\frac{8786824}{11471841}a^{10}-\frac{29862086}{57359205}a^{9}-\frac{17031364}{19119735}a^{8}+\frac{793468}{466335}a^{7}+\frac{162021521}{172077615}a^{6}-\frac{43213528}{172077615}a^{5}-\frac{142178329}{172077615}a^{4}-\frac{6239621}{6373245}a^{3}+\frac{23612012}{34415523}a^{2}+\frac{251518768}{172077615}a+\frac{33743434}{172077615}$, $a$
| 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: | \( 2724154.30052 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 2724154.30052 \cdot 1}{2\cdot\sqrt{1733003264116942402576542823}}\cr\approx \mathstrut & 0.998736283786 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^19 + 9*x^17 - 10*x^16 + 8*x^15 - 45*x^14 + 39*x^13 + 7*x^12 + 42*x^11 - 36*x^10 - 60*x^9 + 9*x^8 + 55*x^7 + 63*x^6 + 15*x^5 - 88*x^4 - 7*x^3 + 27*x^2 + 51*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 + 9*x^17 - 10*x^16 + 8*x^15 - 45*x^14 + 39*x^13 + 7*x^12 + 42*x^11 - 36*x^10 - 60*x^9 + 9*x^8 + 55*x^7 + 63*x^6 + 15*x^5 - 88*x^4 - 7*x^3 + 27*x^2 + 51*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 + 9*x^17 - 10*x^16 + 8*x^15 - 45*x^14 + 39*x^13 + 7*x^12 + 42*x^11 - 36*x^10 - 60*x^9 + 9*x^8 + 55*x^7 + 63*x^6 + 15*x^5 - 88*x^4 - 7*x^3 + 27*x^2 + 51*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 + 9*x^17 - 10*x^16 + 8*x^15 - 45*x^14 + 39*x^13 + 7*x^12 + 42*x^11 - 36*x^10 - 60*x^9 + 9*x^8 + 55*x^7 + 63*x^6 + 15*x^5 - 88*x^4 - 7*x^3 + 27*x^2 + 51*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 | $19$ | ${\href{/padicField/3.2.0.1}{2} }^{9}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.2.0.1}{2} }^{9}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $19$ | $19$ | $19$ | $19$ | $19$ | $19$ | ${\href{/padicField/29.2.0.1}{2} }^{9}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{9}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $19$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19$ | ${\href{/padicField/53.2.0.1}{2} }^{9}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{9}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(1063\)
| $\Q_{1063}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |