Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 5*x^18 + 10*x^17 - 5*x^16 + 9*x^15 - 42*x^14 + 51*x^13 - 6*x^12 - 23*x^11 + 4*x^10 - 29*x^9 + 112*x^8 - 63*x^7 - 147*x^6 + 183*x^5 + 15*x^4 - 77*x^3 - 17*x^2 + 37*x + 1)
gp: K = bnfinit(y^19 - 5*y^18 + 10*y^17 - 5*y^16 + 9*y^15 - 42*y^14 + 51*y^13 - 6*y^12 - 23*y^11 + 4*y^10 - 29*y^9 + 112*y^8 - 63*y^7 - 147*y^6 + 183*y^5 + 15*y^4 - 77*y^3 - 17*y^2 + 37*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 5*x^18 + 10*x^17 - 5*x^16 + 9*x^15 - 42*x^14 + 51*x^13 - 6*x^12 - 23*x^11 + 4*x^10 - 29*x^9 + 112*x^8 - 63*x^7 - 147*x^6 + 183*x^5 + 15*x^4 - 77*x^3 - 17*x^2 + 37*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 5*x^18 + 10*x^17 - 5*x^16 + 9*x^15 - 42*x^14 + 51*x^13 - 6*x^12 - 23*x^11 + 4*x^10 - 29*x^9 + 112*x^8 - 63*x^7 - 147*x^6 + 183*x^5 + 15*x^4 - 77*x^3 - 17*x^2 + 37*x + 1)
\( x^{19} - 5 x^{18} + 10 x^{17} - 5 x^{16} + 9 x^{15} - 42 x^{14} + 51 x^{13} - 6 x^{12} - 23 x^{11} + \cdots + 1 \)
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: |
\(-467562425055097089773569879\)
\(\medspace = -\,919^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.33\) | 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: | $919^{1/2}\approx 30.315012782448235$ | ||
| Ramified primes: |
\(919\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-919}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{14}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{2}{9}a^{7}+\frac{2}{9}a^{6}+\frac{2}{9}a^{5}+\frac{2}{9}a^{4}+\frac{2}{9}a^{3}+\frac{2}{9}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{63}a^{17}+\frac{1}{63}a^{16}-\frac{1}{63}a^{15}-\frac{1}{9}a^{14}+\frac{5}{63}a^{13}+\frac{2}{63}a^{12}-\frac{1}{63}a^{11}-\frac{4}{63}a^{10}-\frac{1}{7}a^{9}-\frac{1}{21}a^{8}-\frac{20}{63}a^{7}+\frac{22}{63}a^{6}+\frac{10}{63}a^{5}+\frac{22}{63}a^{4}-\frac{2}{63}a^{3}+\frac{4}{9}a^{2}-\frac{22}{63}a+\frac{26}{63}$, $\frac{1}{1681146971469}a^{18}-\frac{12482114905}{1681146971469}a^{17}-\frac{26042434111}{560382323823}a^{16}+\frac{1134363586}{88481419551}a^{15}-\frac{142234021442}{1681146971469}a^{14}+\frac{119912754793}{1681146971469}a^{13}+\frac{198152955184}{1681146971469}a^{12}+\frac{230017054447}{1681146971469}a^{11}+\frac{53305188299}{1681146971469}a^{10}-\frac{15535359472}{186794107941}a^{9}+\frac{11712170908}{240163853067}a^{8}+\frac{16274440718}{88481419551}a^{7}+\frac{814706226419}{1681146971469}a^{6}-\frac{36869152762}{88481419551}a^{5}-\frac{418133673766}{1681146971469}a^{4}-\frac{396229955110}{1681146971469}a^{3}-\frac{174508750715}{560382323823}a^{2}+\frac{66551965369}{240163853067}a-\frac{310401455248}{1681146971469}$
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: |
$\frac{19319542178}{560382323823}a^{18}-\frac{69483020147}{560382323823}a^{17}+\frac{12099012913}{80054617689}a^{16}+\frac{4362360565}{29493806517}a^{15}+\frac{120964954798}{560382323823}a^{14}-\frac{54660548189}{80054617689}a^{13}+\frac{28371479002}{560382323823}a^{12}+\frac{604326590950}{560382323823}a^{11}-\frac{85591400178}{62264702647}a^{10}+\frac{555175514192}{560382323823}a^{9}-\frac{970324809529}{560382323823}a^{8}+\frac{60720010541}{29493806517}a^{7}+\frac{1050500602406}{560382323823}a^{6}-\frac{163239325471}{29493806517}a^{5}+\frac{1793772988358}{560382323823}a^{4}+\frac{896598919736}{560382323823}a^{3}-\frac{1618889035511}{560382323823}a^{2}+\frac{535438646471}{186794107941}a-\frac{92475819169}{62264702647}$, $\frac{31371385021}{1681146971469}a^{18}-\frac{154494778117}{1681146971469}a^{17}+\frac{121858645591}{560382323823}a^{16}-\frac{19875218684}{88481419551}a^{15}+\frac{599719357600}{1681146971469}a^{14}-\frac{1147448093585}{1681146971469}a^{13}+\frac{302579301886}{240163853067}a^{12}-\frac{2015185738283}{1681146971469}a^{11}-\frac{66822787768}{1681146971469}a^{10}+\frac{314630136062}{560382323823}a^{9}-\frac{1829036473826}{1681146971469}a^{8}+\frac{183899948894}{88481419551}a^{7}-\frac{498166056808}{240163853067}a^{6}-\frac{34779189583}{88481419551}a^{5}+\frac{6458031278882}{1681146971469}a^{4}-\frac{5539994046997}{1681146971469}a^{3}-\frac{134219627810}{186794107941}a^{2}+\frac{1953017817580}{1681146971469}a+\frac{2013577101266}{1681146971469}$, $\frac{60635750449}{1681146971469}a^{18}-\frac{278425479643}{1681146971469}a^{17}+\frac{62077652008}{186794107941}a^{16}-\frac{14104133339}{88481419551}a^{15}+\frac{639680700265}{1681146971469}a^{14}-\frac{2071590800549}{1681146971469}a^{13}+\frac{3224509064131}{1681146971469}a^{12}-\frac{1302069263}{240163853067}a^{11}-\frac{172919541511}{240163853067}a^{10}+\frac{39263352433}{560382323823}a^{9}-\frac{2038426108181}{1681146971469}a^{8}+\frac{323249445566}{88481419551}a^{7}-\frac{4218001657213}{1681146971469}a^{6}-\frac{435596630887}{88481419551}a^{5}+\frac{10444333500980}{1681146971469}a^{4}+\frac{1641323153009}{1681146971469}a^{3}-\frac{1609164316933}{560382323823}a^{2}-\frac{2441632582604}{1681146971469}a+\frac{1013431653578}{1681146971469}$, $\frac{39150398017}{1681146971469}a^{18}-\frac{226443303097}{1681146971469}a^{17}+\frac{179539836769}{560382323823}a^{16}-\frac{3229432415}{12640202793}a^{15}+\frac{273281689171}{1681146971469}a^{14}-\frac{1595044900919}{1681146971469}a^{13}+\frac{3138122104765}{1681146971469}a^{12}-\frac{996716121287}{1681146971469}a^{11}-\frac{2323870036144}{1681146971469}a^{10}+\frac{740113847456}{560382323823}a^{9}-\frac{1476473857058}{1681146971469}a^{8}+\frac{234917089619}{88481419551}a^{7}-\frac{6387865947790}{1681146971469}a^{6}-\frac{321325943314}{88481419551}a^{5}+\frac{16699378226189}{1681146971469}a^{4}-\frac{639171990283}{240163853067}a^{3}-\frac{1035962591431}{186794107941}a^{2}+\frac{4166723064631}{1681146971469}a+\frac{270088816418}{240163853067}$, $\frac{20182062160}{1681146971469}a^{18}-\frac{141752453890}{1681146971469}a^{17}+\frac{12205326983}{80054617689}a^{16}-\frac{631870802}{88481419551}a^{15}-\frac{135803881166}{1681146971469}a^{14}-\frac{255808423238}{240163853067}a^{13}+\frac{889920344818}{1681146971469}a^{12}+\frac{1450810358314}{1681146971469}a^{11}+\frac{315080864267}{1681146971469}a^{10}-\frac{147209494961}{560382323823}a^{9}-\frac{278841716771}{1681146971469}a^{8}+\frac{203522673458}{88481419551}a^{7}-\frac{114082882966}{1681146971469}a^{6}-\frac{452592882079}{88481419551}a^{5}-\frac{73165204204}{1681146971469}a^{4}+\frac{5764772987825}{1681146971469}a^{3}+\frac{595560840401}{560382323823}a^{2}-\frac{3531428229101}{1681146971469}a-\frac{1667683307131}{1681146971469}$, $\frac{59847194228}{1681146971469}a^{18}-\frac{281882957276}{1681146971469}a^{17}+\frac{169151699296}{560382323823}a^{16}-\frac{986954965}{12640202793}a^{15}+\frac{507715333238}{1681146971469}a^{14}-\frac{2443957375882}{1681146971469}a^{13}+\frac{2189037750008}{1681146971469}a^{12}+\frac{355545056519}{1681146971469}a^{11}-\frac{747273776213}{1681146971469}a^{10}+\frac{30765020843}{560382323823}a^{9}-\frac{2064859245754}{1681146971469}a^{8}+\frac{279782907175}{88481419551}a^{7}-\frac{1355525174498}{1681146971469}a^{6}-\frac{479325975938}{88481419551}a^{5}+\frac{5677740221518}{1681146971469}a^{4}+\frac{359606683756}{240163853067}a^{3}+\frac{54475195774}{560382323823}a^{2}-\frac{1307025971719}{1681146971469}a-\frac{130947599165}{240163853067}$, $\frac{244462039}{17331412077}a^{18}-\frac{1639014691}{17331412077}a^{17}+\frac{1563553666}{5777137359}a^{16}-\frac{336597353}{912179583}a^{15}+\frac{6802442851}{17331412077}a^{14}-\frac{16948780874}{17331412077}a^{13}+\frac{4813625533}{2475916011}a^{12}-\frac{34029151796}{17331412077}a^{11}+\frac{12699830936}{17331412077}a^{10}-\frac{1636234289}{5777137359}a^{9}-\frac{419765498}{17331412077}a^{8}+\frac{1775899352}{912179583}a^{7}-\frac{9460383745}{2475916011}a^{6}+\frac{844263803}{912179583}a^{5}+\frac{79620539438}{17331412077}a^{4}-\frac{61886457076}{17331412077}a^{3}-\frac{1303300412}{5777137359}a^{2}+\frac{4650098701}{17331412077}a+\frac{16970468687}{17331412077}$, $\frac{22180601329}{1681146971469}a^{18}-\frac{46385971783}{1681146971469}a^{17}+\frac{2128912871}{80054617689}a^{16}-\frac{1685300051}{88481419551}a^{15}+\frac{685919911744}{1681146971469}a^{14}-\frac{21597646931}{240163853067}a^{13}-\frac{7938937379}{1681146971469}a^{12}-\frac{1411345656410}{1681146971469}a^{11}+\frac{1152402719732}{1681146971469}a^{10}+\frac{6964977356}{560382323823}a^{9}-\frac{1245677731217}{1681146971469}a^{8}-\frac{10386481621}{88481419551}a^{7}+\frac{879719857400}{1681146971469}a^{6}+\frac{148418284676}{88481419551}a^{5}-\frac{1061257916668}{1681146971469}a^{4}-\frac{5015940806152}{1681146971469}a^{3}+\frac{1120848423173}{560382323823}a^{2}+\frac{2497906017004}{1681146971469}a+\frac{391203203990}{1681146971469}$, $a$
| 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: | \( 1152808.1284 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 1152808.1284 \cdot 1}{2\cdot\sqrt{467562425055097089773569879}}\cr\approx \mathstrut & 0.81368450161 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^19 - 5*x^18 + 10*x^17 - 5*x^16 + 9*x^15 - 42*x^14 + 51*x^13 - 6*x^12 - 23*x^11 + 4*x^10 - 29*x^9 + 112*x^8 - 63*x^7 - 147*x^6 + 183*x^5 + 15*x^4 - 77*x^3 - 17*x^2 + 37*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 - 5*x^18 + 10*x^17 - 5*x^16 + 9*x^15 - 42*x^14 + 51*x^13 - 6*x^12 - 23*x^11 + 4*x^10 - 29*x^9 + 112*x^8 - 63*x^7 - 147*x^6 + 183*x^5 + 15*x^4 - 77*x^3 - 17*x^2 + 37*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 - 5*x^18 + 10*x^17 - 5*x^16 + 9*x^15 - 42*x^14 + 51*x^13 - 6*x^12 - 23*x^11 + 4*x^10 - 29*x^9 + 112*x^8 - 63*x^7 - 147*x^6 + 183*x^5 + 15*x^4 - 77*x^3 - 17*x^2 + 37*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 - 5*x^18 + 10*x^17 - 5*x^16 + 9*x^15 - 42*x^14 + 51*x^13 - 6*x^12 - 23*x^11 + 4*x^10 - 29*x^9 + 112*x^8 - 63*x^7 - 147*x^6 + 183*x^5 + 15*x^4 - 77*x^3 - 17*x^2 + 37*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 | $19$ | ${\href{/padicField/3.2.0.1}{2} }^{9}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $19$ | ${\href{/padicField/7.2.0.1}{2} }^{9}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $19$ | $19$ | $19$ | ${\href{/padicField/19.2.0.1}{2} }^{9}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $19$ | $19$ | ${\href{/padicField/31.2.0.1}{2} }^{9}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{9}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $19$ | ${\href{/padicField/53.2.0.1}{2} }^{9}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $19$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(919\)
| $\Q_{919}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |