Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 2*x^18 - 18*x^17 + 32*x^16 + 132*x^15 - 202*x^14 - 502*x^13 + 645*x^12 + 1045*x^11 - 1122*x^10 - 1176*x^9 + 1078*x^8 + 700*x^7 - 557*x^6 - 220*x^5 + 151*x^4 + 34*x^3 - 20*x^2 - 2*x + 1)
gp: K = bnfinit(y^19 - 2*y^18 - 18*y^17 + 32*y^16 + 132*y^15 - 202*y^14 - 502*y^13 + 645*y^12 + 1045*y^11 - 1122*y^10 - 1176*y^9 + 1078*y^8 + 700*y^7 - 557*y^6 - 220*y^5 + 151*y^4 + 34*y^3 - 20*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 2*x^18 - 18*x^17 + 32*x^16 + 132*x^15 - 202*x^14 - 502*x^13 + 645*x^12 + 1045*x^11 - 1122*x^10 - 1176*x^9 + 1078*x^8 + 700*x^7 - 557*x^6 - 220*x^5 + 151*x^4 + 34*x^3 - 20*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 2*x^18 - 18*x^17 + 32*x^16 + 132*x^15 - 202*x^14 - 502*x^13 + 645*x^12 + 1045*x^11 - 1122*x^10 - 1176*x^9 + 1078*x^8 + 700*x^7 - 557*x^6 - 220*x^5 + 151*x^4 + 34*x^3 - 20*x^2 - 2*x + 1)
\( x^{19} - 2 x^{18} - 18 x^{17} + 32 x^{16} + 132 x^{15} - 202 x^{14} - 502 x^{13} + 645 x^{12} + 1045 x^{11} + \cdots + 1 \)
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: | $[13, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-801631882964189735135508990628664\)
\(\medspace = -\,2^{3}\cdot 13^{2}\cdot 17\cdot 3229\cdot 34365587\cdot 85132367\cdot 3692009831\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(53.92\) | 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: | $2^{3/2}13^{2/3}17^{1/2}3229^{1/2}34365587^{1/2}85132367^{1/2}3692009831^{1/2}\approx 1.2041290670651694e+16$ | ||
| Ramified primes: |
\(2\), \(13\), \(17\), \(3229\), \(34365587\), \(85132367\), \(3692009831\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-11858\!\cdots\!09214}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $15$ | 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: |
$a$, $a^{17}-2a^{16}-18a^{15}+32a^{14}+132a^{13}-202a^{12}-502a^{11}+645a^{10}+1045a^{9}-1122a^{8}-1176a^{7}+1078a^{6}+700a^{5}-557a^{4}-219a^{3}+150a^{2}+29a-17$, $51a^{18}-78a^{17}-955a^{16}+1184a^{15}+7293a^{14}-6894a^{13}-28863a^{12}+19476a^{11}+62481a^{10}-28386a^{9}-73289a^{8}+21531a^{7}+45679a^{6}-7882a^{5}-14779a^{4}+1164a^{3}+2231a^{2}-48a-119$, $2a^{18}-3a^{17}-38a^{16}+46a^{15}+296a^{14}-272a^{13}-1206a^{12}+788a^{11}+2735a^{10}-1199a^{9}-3474a^{8}+980a^{7}+2478a^{6}-414a^{5}-996a^{4}+82a^{3}+213a^{2}-7a-18$, $14a^{18}-49a^{17}-205a^{16}+810a^{15}+1100a^{14}-5338a^{13}-2349a^{12}+17867a^{11}-28a^{10}-32037a^{9}+7939a^{8}+29913a^{7}-11415a^{6}-13498a^{5}+5791a^{4}+2813a^{3}-1180a^{2}-214a+81$, $36a^{18}-50a^{17}-685a^{16}+747a^{15}+5321a^{14}-4235a^{13}-21452a^{12}+11436a^{11}+47450a^{10}-15467a^{9}-57184a^{8}+10452a^{7}+36759a^{6}-3084a^{5}-12132a^{4}+173a^{3}+1836a^{2}+29a-97$, $37a^{18}-88a^{17}-628a^{16}+1412a^{15}+4267a^{14}-8937a^{13}-14605a^{12}+28457a^{11}+25769a^{10}-48391a^{9}-21110a^{8}+43175a^{7}+5575a^{6}-18663a^{5}+746a^{4}+3590a^{3}-475a^{2}-239a+43$, $83a^{18}-128a^{17}-1560a^{16}+1955a^{15}+11983a^{14}-11481a^{13}-47870a^{12}+32798a^{11}+105202a^{10}-48332a^{9}-126315a^{8}+36627a^{7}+81059a^{6}-12924a^{5}-26779a^{4}+1657a^{3}+4052a^{2}-36a-212$, $142a^{18}-188a^{17}-2718a^{16}+2778a^{15}+21238a^{14}-15455a^{13}-86129a^{12}+40343a^{11}+191738a^{10}-51236a^{9}-233022a^{8}+30800a^{7}+151717a^{6}-6790a^{5}-50997a^{4}-480a^{3}+7876a^{2}+196a-424$, $105a^{18}-180a^{17}-1927a^{16}+2780a^{15}+14399a^{14}-16631a^{13}-55633a^{12}+48951a^{11}+116991a^{10}-75483a^{9}-131816a^{8}+60898a^{7}+77439a^{6}-23702a^{5}-23241a^{4}+3881a^{3}+3170a^{2}-199a-143$, $98a^{18}-239a^{17}-1646a^{16}+3830a^{15}+11027a^{14}-24188a^{13}-36980a^{12}+76670a^{11}+63037a^{10}-129072a^{9}-47762a^{8}+112671a^{7}+8811a^{6}-46711a^{5}+3511a^{4}+8593a^{3}-1373a^{2}-552a+116$, $4a^{18}-14a^{17}-58a^{16}+231a^{15}+303a^{14}-1520a^{13}-581a^{12}+5086a^{11}-376a^{10}-9141a^{9}+3075a^{8}+8587a^{7}-4199a^{6}-3890a^{5}+2213a^{4}+792a^{3}-480a^{2}-56a+35$, $117a^{18}-302a^{17}-1936a^{16}+4883a^{15}+12701a^{14}-31241a^{13}-41241a^{12}+100886a^{11}+66039a^{10}-174352a^{9}-41026a^{8}+157987a^{7}-4958a^{6}-69302a^{5}+12070a^{4}+13596a^{3}-3358a^{2}-926a+258$, $4a^{18}+20a^{17}-128a^{16}-369a^{15}+1410a^{14}+2764a^{13}-7444a^{12}-10587a^{11}+20888a^{10}+21506a^{9}-31980a^{8}-22203a^{7}+26257a^{6}+10899a^{5}-10722a^{4}-2427a^{3}+1964a^{2}+189a-124$, $47a^{18}-91a^{17}-841a^{16}+1428a^{15}+6105a^{14}-8752a^{13}-22801a^{12}+26681a^{11}+45906a^{10}-43079a^{9}-48518a^{8}+36546a^{7}+25849a^{6}-15047a^{5}-6905a^{4}+2715a^{3}+837a^{2}-166a-35$
| 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: | \( 23798894722.8 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{3}\cdot 23798894722.8 \cdot 1}{2\cdot\sqrt{801631882964189735135508990628664}}\cr\approx \mathstrut & 0.854021589998 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^19 - 2*x^18 - 18*x^17 + 32*x^16 + 132*x^15 - 202*x^14 - 502*x^13 + 645*x^12 + 1045*x^11 - 1122*x^10 - 1176*x^9 + 1078*x^8 + 700*x^7 - 557*x^6 - 220*x^5 + 151*x^4 + 34*x^3 - 20*x^2 - 2*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 - 18*x^17 + 32*x^16 + 132*x^15 - 202*x^14 - 502*x^13 + 645*x^12 + 1045*x^11 - 1122*x^10 - 1176*x^9 + 1078*x^8 + 700*x^7 - 557*x^6 - 220*x^5 + 151*x^4 + 34*x^3 - 20*x^2 - 2*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 - 18*x^17 + 32*x^16 + 132*x^15 - 202*x^14 - 502*x^13 + 645*x^12 + 1045*x^11 - 1122*x^10 - 1176*x^9 + 1078*x^8 + 700*x^7 - 557*x^6 - 220*x^5 + 151*x^4 + 34*x^3 - 20*x^2 - 2*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 - 18*x^17 + 32*x^16 + 132*x^15 - 202*x^14 - 502*x^13 + 645*x^12 + 1045*x^11 - 1122*x^10 - 1176*x^9 + 1078*x^8 + 700*x^7 - 557*x^6 - 220*x^5 + 151*x^4 + 34*x^3 - 20*x^2 - 2*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 non-solvable group of order 121645100408832000 |
| The 490 conjugacy class representatives for $S_{19}$ |
| Character table for $S_{19}$ |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $19$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ | $19$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | R | $19$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | $17{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | $19$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
In the table, R denotes a ramified prime. 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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x^{2} + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(13\)
| 13.3.2.3 | $x^{3} + 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.6.0.1 | $x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(3229\)
| $\Q_{3229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(34365587\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(85132367\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
|
\(3692009831\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |