Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 42*x^15 - 167*x^12 + 13136*x^9 - 110969*x^6 + 5921929*x^3 + 128210329)
gp: K = bnfinit(y^18 - 42*y^15 - 167*y^12 + 13136*y^9 - 110969*y^6 + 5921929*y^3 + 128210329, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 42*x^15 - 167*x^12 + 13136*x^9 - 110969*x^6 + 5921929*x^3 + 128210329);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 42*x^15 - 167*x^12 + 13136*x^9 - 110969*x^6 + 5921929*x^3 + 128210329)
\( x^{18} - 42x^{15} - 167x^{12} + 13136x^{9} - 110969x^{6} + 5921929x^{3} + 128210329 \)
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: |
\(-206475298754977298051321054411907\)
\(\medspace = -\,3^{27}\cdot 13^{4}\cdot 19^{6}\cdot 67^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(62.41\) | 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: |
\(3\), \(13\), \(19\), \(67\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13}a^{14}-\frac{3}{13}a^{11}+\frac{2}{13}a^{8}+\frac{6}{13}a^{5}-\frac{1}{13}a^{2}$, $\frac{1}{32\!\cdots\!69}a^{15}-\frac{12\!\cdots\!53}{32\!\cdots\!69}a^{12}-\frac{13\!\cdots\!17}{32\!\cdots\!69}a^{9}+\frac{10\!\cdots\!39}{32\!\cdots\!69}a^{6}-\frac{51\!\cdots\!77}{32\!\cdots\!69}a^{3}+\frac{583315130780949}{29\!\cdots\!03}$, $\frac{1}{32\!\cdots\!69}a^{16}-\frac{12\!\cdots\!53}{32\!\cdots\!69}a^{13}-\frac{13\!\cdots\!17}{32\!\cdots\!69}a^{10}+\frac{10\!\cdots\!39}{32\!\cdots\!69}a^{7}-\frac{51\!\cdots\!77}{32\!\cdots\!69}a^{4}+\frac{583315130780949}{29\!\cdots\!03}a$, $\frac{1}{42\!\cdots\!97}a^{17}-\frac{12\!\cdots\!53}{42\!\cdots\!97}a^{14}+\frac{11\!\cdots\!59}{42\!\cdots\!97}a^{11}-\frac{55\!\cdots\!99}{42\!\cdots\!97}a^{8}-\frac{13\!\cdots\!53}{42\!\cdots\!97}a^{5}-\frac{52\!\cdots\!57}{37\!\cdots\!39}a^{2}$
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
$C_{3}$, which has order $3$ (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: |
\( \frac{284411}{282429367867} a^{15} - \frac{15830698}{282429367867} a^{12} + \frac{198562921}{282429367867} a^{9} + \frac{86957229}{282429367867} a^{6} - \frac{25223556397}{282429367867} a^{3} + \frac{2163792571721}{282429367867} \)
(order $6$)
| 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: |
$\frac{142428287929}{32\!\cdots\!69}a^{15}+\frac{98589636727619}{32\!\cdots\!69}a^{12}-\frac{50\!\cdots\!51}{32\!\cdots\!69}a^{9}+\frac{53\!\cdots\!24}{32\!\cdots\!69}a^{6}-\frac{49\!\cdots\!58}{32\!\cdots\!69}a^{3}-\frac{579313490364413}{29\!\cdots\!03}$, $\frac{23399719505491}{32\!\cdots\!69}a^{15}-\frac{14\!\cdots\!78}{32\!\cdots\!69}a^{12}+\frac{19\!\cdots\!08}{32\!\cdots\!69}a^{9}+\frac{54\!\cdots\!71}{32\!\cdots\!69}a^{6}-\frac{23\!\cdots\!71}{32\!\cdots\!69}a^{3}+\frac{15\!\cdots\!14}{29\!\cdots\!03}$, $\frac{622366899619736}{42\!\cdots\!97}a^{17}-\frac{21353455285831}{32\!\cdots\!69}a^{16}-\frac{376382428075785}{32\!\cdots\!69}a^{15}-\frac{37\!\cdots\!92}{42\!\cdots\!97}a^{14}+\frac{11\!\cdots\!69}{32\!\cdots\!69}a^{13}+\frac{21\!\cdots\!01}{32\!\cdots\!69}a^{12}+\frac{53\!\cdots\!13}{42\!\cdots\!97}a^{11}-\frac{10\!\cdots\!47}{32\!\cdots\!69}a^{10}-\frac{27\!\cdots\!02}{32\!\cdots\!69}a^{9}-\frac{27\!\cdots\!53}{42\!\cdots\!97}a^{8}-\frac{31\!\cdots\!65}{32\!\cdots\!69}a^{7}-\frac{91\!\cdots\!95}{32\!\cdots\!69}a^{6}-\frac{62\!\cdots\!72}{42\!\cdots\!97}a^{5}+\frac{11\!\cdots\!42}{32\!\cdots\!69}a^{4}+\frac{55\!\cdots\!14}{32\!\cdots\!69}a^{3}+\frac{39\!\cdots\!48}{37\!\cdots\!39}a^{2}-\frac{25\!\cdots\!43}{29\!\cdots\!03}a-\frac{26\!\cdots\!14}{29\!\cdots\!03}$, $\frac{250241753593775}{42\!\cdots\!97}a^{17}-\frac{21353455285831}{32\!\cdots\!69}a^{16}-\frac{473158720740358}{32\!\cdots\!69}a^{15}-\frac{13\!\cdots\!33}{42\!\cdots\!97}a^{14}+\frac{11\!\cdots\!69}{32\!\cdots\!69}a^{13}+\frac{28\!\cdots\!02}{32\!\cdots\!69}a^{12}+\frac{14\!\cdots\!68}{42\!\cdots\!97}a^{11}-\frac{10\!\cdots\!47}{32\!\cdots\!69}a^{10}-\frac{42\!\cdots\!32}{32\!\cdots\!69}a^{9}+\frac{86\!\cdots\!73}{42\!\cdots\!97}a^{8}-\frac{31\!\cdots\!65}{32\!\cdots\!69}a^{7}+\frac{11\!\cdots\!97}{32\!\cdots\!69}a^{6}-\frac{28\!\cdots\!07}{42\!\cdots\!97}a^{5}+\frac{11\!\cdots\!42}{32\!\cdots\!69}a^{4}+\frac{43\!\cdots\!79}{32\!\cdots\!69}a^{3}+\frac{18\!\cdots\!42}{37\!\cdots\!39}a^{2}-\frac{25\!\cdots\!43}{29\!\cdots\!03}a-\frac{31\!\cdots\!93}{29\!\cdots\!03}$, $\frac{221159163014320}{42\!\cdots\!97}a^{17}+\frac{40\!\cdots\!97}{32\!\cdots\!69}a^{16}-\frac{141311908212886}{49\!\cdots\!07}a^{15}-\frac{23\!\cdots\!68}{42\!\cdots\!97}a^{14}-\frac{39\!\cdots\!21}{32\!\cdots\!69}a^{13}+\frac{24\!\cdots\!78}{49\!\cdots\!07}a^{12}+\frac{85\!\cdots\!96}{42\!\cdots\!97}a^{11}-\frac{13\!\cdots\!62}{32\!\cdots\!69}a^{10}+\frac{76\!\cdots\!62}{49\!\cdots\!07}a^{9}-\frac{30\!\cdots\!59}{42\!\cdots\!97}a^{8}-\frac{11\!\cdots\!22}{32\!\cdots\!69}a^{7}+\frac{92\!\cdots\!62}{49\!\cdots\!07}a^{6}-\frac{67\!\cdots\!21}{42\!\cdots\!97}a^{5}-\frac{95\!\cdots\!78}{32\!\cdots\!69}a^{4}+\frac{44\!\cdots\!72}{49\!\cdots\!07}a^{3}+\frac{16\!\cdots\!87}{37\!\cdots\!39}a^{2}-\frac{92\!\cdots\!48}{29\!\cdots\!03}a+\frac{29\!\cdots\!30}{29\!\cdots\!03}$, $\frac{221159163014320}{42\!\cdots\!97}a^{17}-\frac{33\!\cdots\!75}{32\!\cdots\!69}a^{16}+\frac{204203096639648}{49\!\cdots\!07}a^{15}-\frac{23\!\cdots\!68}{42\!\cdots\!97}a^{14}+\frac{69\!\cdots\!58}{32\!\cdots\!69}a^{13}-\frac{23\!\cdots\!96}{49\!\cdots\!07}a^{12}+\frac{85\!\cdots\!96}{42\!\cdots\!97}a^{11}+\frac{13\!\cdots\!86}{32\!\cdots\!69}a^{10}-\frac{63\!\cdots\!88}{49\!\cdots\!07}a^{9}-\frac{30\!\cdots\!59}{42\!\cdots\!97}a^{8}+\frac{19\!\cdots\!26}{32\!\cdots\!69}a^{7}-\frac{64\!\cdots\!54}{49\!\cdots\!07}a^{6}-\frac{67\!\cdots\!21}{42\!\cdots\!97}a^{5}+\frac{10\!\cdots\!82}{32\!\cdots\!69}a^{4}-\frac{48\!\cdots\!56}{49\!\cdots\!07}a^{3}+\frac{16\!\cdots\!87}{37\!\cdots\!39}a^{2}+\frac{70\!\cdots\!23}{29\!\cdots\!03}a-\frac{28\!\cdots\!50}{29\!\cdots\!03}$, $\frac{103427683069846}{32\!\cdots\!69}a^{17}+\frac{131366700052431}{32\!\cdots\!69}a^{16}-\frac{7704639542765}{19\!\cdots\!01}a^{15}+\frac{11\!\cdots\!46}{32\!\cdots\!69}a^{14}-\frac{21\!\cdots\!14}{32\!\cdots\!69}a^{13}+\frac{471972530156934}{19\!\cdots\!01}a^{12}-\frac{24\!\cdots\!02}{32\!\cdots\!69}a^{11}+\frac{54\!\cdots\!31}{32\!\cdots\!69}a^{10}-\frac{34\!\cdots\!23}{19\!\cdots\!01}a^{9}-\frac{65\!\cdots\!13}{32\!\cdots\!69}a^{8}+\frac{13\!\cdots\!63}{32\!\cdots\!69}a^{7}-\frac{28\!\cdots\!39}{19\!\cdots\!01}a^{6}+\frac{11\!\cdots\!91}{32\!\cdots\!69}a^{5}-\frac{30\!\cdots\!85}{32\!\cdots\!69}a^{4}+\frac{29\!\cdots\!14}{19\!\cdots\!01}a^{3}+\frac{86\!\cdots\!62}{29\!\cdots\!03}a^{2}-\frac{40\!\cdots\!07}{29\!\cdots\!03}a+\frac{58\!\cdots\!71}{17228191439887}$, $\frac{241886121501356}{32\!\cdots\!69}a^{17}+\frac{716587310858708}{32\!\cdots\!69}a^{16}+\frac{7704639542765}{19\!\cdots\!01}a^{15}-\frac{91\!\cdots\!81}{32\!\cdots\!69}a^{14}-\frac{36\!\cdots\!10}{32\!\cdots\!69}a^{13}-\frac{471972530156934}{19\!\cdots\!01}a^{12}-\frac{17\!\cdots\!71}{32\!\cdots\!69}a^{11}-\frac{16\!\cdots\!80}{32\!\cdots\!69}a^{10}+\frac{34\!\cdots\!23}{19\!\cdots\!01}a^{9}+\frac{55\!\cdots\!92}{32\!\cdots\!69}a^{8}+\frac{22\!\cdots\!66}{32\!\cdots\!69}a^{7}+\frac{28\!\cdots\!39}{19\!\cdots\!01}a^{6}+\frac{64\!\cdots\!09}{32\!\cdots\!69}a^{5}+\frac{62\!\cdots\!80}{32\!\cdots\!69}a^{4}-\frac{29\!\cdots\!14}{19\!\cdots\!01}a^{3}-\frac{19\!\cdots\!13}{29\!\cdots\!03}a^{2}-\frac{18\!\cdots\!03}{29\!\cdots\!03}a-\frac{58\!\cdots\!71}{17228191439887}$
| 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: | \( 495723249.783 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 495723249.783 \cdot 3}{6\cdot\sqrt{206475298754977298051321054411907}}\cr\approx \mathstrut & 0.263265564245 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 42*x^15 - 167*x^12 + 13136*x^9 - 110969*x^6 + 5921929*x^3 + 128210329)
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 - 42*x^15 - 167*x^12 + 13136*x^9 - 110969*x^6 + 5921929*x^3 + 128210329, 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 - 42*x^15 - 167*x^12 + 13136*x^9 - 110969*x^6 + 5921929*x^3 + 128210329);
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 - 42*x^15 - 167*x^12 + 13136*x^9 - 110969*x^6 + 5921929*x^3 + 128210329);
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
$C_3\wr (C_3\times S_3)$ (as 18T584):
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 13122 |
| The 170 conjugacy class representatives for $C_3\wr (C_3\times S_3)$ |
| Character table for $C_3\wr (C_3\times S_3)$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 6.0.9747.1 |
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 |
| Minimal sibling: | 18.0.272164698131233743036984627.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | $18$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | R | $18$ | $18$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $6$ | $3$ | $27$ | |||
|
\(13\)
| 13.3.2.3 | $x^{3} + 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.2.3 | $x^{3} + 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
|
\(67\)
| 67.3.2.2 | $x^{3} + 268$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 67.3.2.3 | $x^{3} + 134$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 67.9.0.1 | $x^{9} + 25 x^{3} + 49 x^{2} + 55 x + 65$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |