Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 48*x^16 + 873*x^14 + 8175*x^12 + 43777*x^10 + 137970*x^8 + 250326*x^6 + 242159*x^4 + 106935*x^2 + 16361)
gp: K = bnfinit(y^18 + 48*y^16 + 873*y^14 + 8175*y^12 + 43777*y^10 + 137970*y^8 + 250326*y^6 + 242159*y^4 + 106935*y^2 + 16361, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 48*x^16 + 873*x^14 + 8175*x^12 + 43777*x^10 + 137970*x^8 + 250326*x^6 + 242159*x^4 + 106935*x^2 + 16361);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 48*x^16 + 873*x^14 + 8175*x^12 + 43777*x^10 + 137970*x^8 + 250326*x^6 + 242159*x^4 + 106935*x^2 + 16361)
\( x^{18} + 48 x^{16} + 873 x^{14} + 8175 x^{12} + 43777 x^{10} + 137970 x^{8} + 250326 x^{6} + \cdots + 16361 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-12065345096686615429034956292096\)
\(\medspace = -\,2^{30}\cdot 37^{6}\cdot 16361^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(53.30\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(37\), \(16361\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-16361}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{15}a^{14}+\frac{1}{3}a^{12}-\frac{1}{5}a^{10}+\frac{4}{15}a^{8}+\frac{1}{5}a^{6}+\frac{2}{15}a^{4}+\frac{7}{15}a^{2}-\frac{4}{15}$, $\frac{1}{15}a^{15}+\frac{1}{3}a^{13}-\frac{1}{5}a^{11}+\frac{4}{15}a^{9}+\frac{1}{5}a^{7}+\frac{2}{15}a^{5}+\frac{7}{15}a^{3}-\frac{4}{15}a$, $\frac{1}{67768010415675}a^{16}+\frac{1009732885009}{67768010415675}a^{14}-\frac{14458417267328}{67768010415675}a^{12}+\frac{18361164407317}{67768010415675}a^{10}-\frac{12484710804661}{67768010415675}a^{8}+\frac{12058067964149}{67768010415675}a^{6}-\frac{217199827232}{13553602083135}a^{4}+\frac{19219104522574}{67768010415675}a^{2}+\frac{30388774912774}{67768010415675}$, $\frac{1}{67768010415675}a^{17}+\frac{1009732885009}{67768010415675}a^{15}-\frac{14458417267328}{67768010415675}a^{13}+\frac{18361164407317}{67768010415675}a^{11}-\frac{12484710804661}{67768010415675}a^{9}+\frac{12058067964149}{67768010415675}a^{7}-\frac{217199827232}{13553602083135}a^{5}+\frac{19219104522574}{67768010415675}a^{3}+\frac{30388774912774}{67768010415675}a$
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
$C_{4}\times C_{560}$, which has order $2240$ (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: | $8$ | 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: |
$\frac{79070311784}{67768010415675}a^{16}+\frac{3635980474391}{67768010415675}a^{14}+\frac{61441576826948}{67768010415675}a^{12}+\frac{512257581840173}{67768010415675}a^{10}+\frac{22\!\cdots\!91}{67768010415675}a^{8}+\frac{54\!\cdots\!21}{67768010415675}a^{6}+\frac{12\!\cdots\!51}{13553602083135}a^{4}+\frac{28\!\cdots\!61}{67768010415675}a^{2}+\frac{341187448050101}{67768010415675}$, $\frac{94242087358}{67768010415675}a^{16}+\frac{4126667413957}{67768010415675}a^{14}+\frac{65101432428076}{67768010415675}a^{12}+\frac{505440039058906}{67768010415675}a^{10}+\frac{21\!\cdots\!27}{67768010415675}a^{8}+\frac{51\!\cdots\!22}{67768010415675}a^{6}+\frac{13\!\cdots\!58}{13553602083135}a^{4}+\frac{46\!\cdots\!12}{67768010415675}a^{2}+\frac{11\!\cdots\!27}{67768010415675}$, $\frac{158046191}{63040009689}a^{16}+\frac{7030486547}{63040009689}a^{14}+\frac{113572282286}{63040009689}a^{12}+\frac{906099927095}{63040009689}a^{10}+\frac{3930192713938}{63040009689}a^{8}+\frac{9361453411909}{63040009689}a^{6}+\frac{11553775530055}{63040009689}a^{4}+\frac{6373777051313}{63040009689}a^{2}+\frac{1110172541054}{63040009689}$, $\frac{230815554173}{67768010415675}a^{16}+\frac{10128743248802}{67768010415675}a^{14}+\frac{159662431397381}{67768010415675}a^{12}+\frac{12\!\cdots\!56}{67768010415675}a^{10}+\frac{49\!\cdots\!77}{67768010415675}a^{8}+\frac{10\!\cdots\!37}{67768010415675}a^{6}+\frac{22\!\cdots\!77}{13553602083135}a^{4}+\frac{49\!\cdots\!67}{67768010415675}a^{2}+\frac{694823389770347}{67768010415675}$, $\frac{30276447847}{22589336805225}a^{16}+\frac{1307264187878}{22589336805225}a^{14}+\frac{6738736292278}{7529778935075}a^{12}+\frac{153933279928984}{22589336805225}a^{10}+\frac{214970678448351}{7529778935075}a^{8}+\frac{15\!\cdots\!18}{22589336805225}a^{6}+\frac{410649419515958}{4517867361045}a^{4}+\frac{13\!\cdots\!38}{22589336805225}a^{2}+\frac{94694225953661}{7529778935075}$, $\frac{2204983284}{7529778935075}a^{16}+\frac{314417905393}{22589336805225}a^{14}+\frac{5581082702069}{22589336805225}a^{12}+\frac{16529581764878}{7529778935075}a^{10}+\frac{240650605237303}{22589336805225}a^{8}+\frac{212393634046116}{7529778935075}a^{6}+\frac{169756213639141}{4517867361045}a^{4}+\frac{474674668552723}{22589336805225}a^{2}+\frac{111147571356323}{22589336805225}$, $\frac{271898479933}{67768010415675}a^{16}+\frac{12091751322277}{67768010415675}a^{14}+\frac{195018583517026}{67768010415675}a^{12}+\frac{15\!\cdots\!71}{67768010415675}a^{10}+\frac{66\!\cdots\!32}{67768010415675}a^{8}+\frac{15\!\cdots\!07}{67768010415675}a^{6}+\frac{38\!\cdots\!86}{13553602083135}a^{4}+\frac{11\!\cdots\!52}{67768010415675}a^{2}+\frac{21\!\cdots\!72}{67768010415675}$, $\frac{75657567967}{67768010415675}a^{16}+\frac{3431105624068}{67768010415675}a^{14}+\frac{56988771029374}{67768010415675}a^{12}+\frac{468617382568219}{67768010415675}a^{10}+\frac{20\!\cdots\!23}{67768010415675}a^{8}+\frac{49\!\cdots\!53}{67768010415675}a^{6}+\frac{11\!\cdots\!67}{13553602083135}a^{4}+\frac{22\!\cdots\!63}{67768010415675}a^{2}+\frac{38784813339223}{67768010415675}$
| 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: | \( 45363.6836572 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 45363.6836572 \cdot 2240}{2\cdot\sqrt{12065345096686615429034956292096}}\cr\approx \mathstrut & 0.223241677366 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 48*x^16 + 873*x^14 + 8175*x^12 + 43777*x^10 + 137970*x^8 + 250326*x^6 + 242159*x^4 + 106935*x^2 + 16361)
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^18 + 48*x^16 + 873*x^14 + 8175*x^12 + 43777*x^10 + 137970*x^8 + 250326*x^6 + 242159*x^4 + 106935*x^2 + 16361, 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^18 + 48*x^16 + 873*x^14 + 8175*x^12 + 43777*x^10 + 137970*x^8 + 250326*x^6 + 242159*x^4 + 106935*x^2 + 16361);
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^18 + 48*x^16 + 873*x^14 + 8175*x^12 + 43777*x^10 + 137970*x^8 + 250326*x^6 + 242159*x^4 + 106935*x^2 + 16361);
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
$S_4^3.D_6$ (as 18T837):
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 165888 |
| The 130 conjugacy class representatives for $S_4^3.D_6$ |
| Character table for $S_4^3.D_6$ |
Intermediate fields
| 3.3.148.1, 9.9.53038958912.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.0.737445455454227457309147136.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | $18$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.30.57 | $x^{18} + 2 x^{17} + 26 x^{16} + 88 x^{15} + 282 x^{14} + 560 x^{13} + 1082 x^{12} + 1648 x^{11} + 2616 x^{10} + 3024 x^{9} + 3768 x^{8} + 3544 x^{7} + 4084 x^{6} + 3048 x^{5} + 2888 x^{4} + 1552 x^{3} + 1544 x^{2} + 592 x + 344$ | $6$ | $3$ | $30$ | 18T366 | $[2, 2, 2, 8/3, 8/3, 8/3, 8/3]_{3}^{6}$ |
|
\(37\)
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.8.4.1 | $x^{8} + 3700 x^{7} + 5133910 x^{6} + 3166256548 x^{5} + 732510094073 x^{4} + 136269235536 x^{3} + 4476368972260 x^{2} + 17928293629116 x + 2173698901413$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(16361\)
| $\Q_{16361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{16361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{16361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{16361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |