Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 57*x^15 + 954*x^12 - 3931*x^9 + 6396*x^6 - 4563*x^3 + 2197)
gp: K = bnfinit(y^18 - 57*y^15 + 954*y^12 - 3931*y^9 + 6396*y^6 - 4563*y^3 + 2197, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 57*x^15 + 954*x^12 - 3931*x^9 + 6396*x^6 - 4563*x^3 + 2197);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 57*x^15 + 954*x^12 - 3931*x^9 + 6396*x^6 - 4563*x^3 + 2197)
\( x^{18} - 57x^{15} + 954x^{12} - 3931x^{9} + 6396x^{6} - 4563x^{3} + 2197 \)
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: |
\(-569752789846542015713684226048\)
\(\medspace = -\,2^{18}\cdot 3^{37}\cdot 13^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(44.99\) | 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: | $2^{3/2}3^{37/18}13^{2/3}\approx 149.59708836062785$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{2}$, $\frac{1}{78}a^{12}+\frac{4}{39}a^{9}-\frac{7}{26}a^{6}+\frac{17}{39}a^{3}-\frac{1}{6}$, $\frac{1}{78}a^{13}+\frac{4}{39}a^{10}-\frac{7}{26}a^{7}+\frac{17}{39}a^{4}-\frac{1}{6}a$, $\frac{1}{234}a^{14}-\frac{1}{234}a^{13}+\frac{1}{234}a^{12}+\frac{17}{117}a^{11}-\frac{17}{117}a^{10}+\frac{17}{117}a^{9}+\frac{19}{78}a^{8}-\frac{19}{78}a^{7}+\frac{19}{78}a^{6}-\frac{22}{117}a^{5}+\frac{22}{117}a^{4}-\frac{22}{117}a^{3}+\frac{1}{18}a^{2}-\frac{1}{18}a+\frac{1}{18}$, $\frac{1}{248826474}a^{15}+\frac{776260}{124413237}a^{12}-\frac{16679723}{248826474}a^{9}+\frac{30490724}{124413237}a^{6}+\frac{3434933}{19140498}a^{3}+\frac{86054}{736173}$, $\frac{1}{248826474}a^{16}+\frac{776260}{124413237}a^{13}-\frac{16679723}{248826474}a^{10}+\frac{30490724}{124413237}a^{7}+\frac{3434933}{19140498}a^{4}+\frac{86054}{736173}a$, $\frac{1}{248826474}a^{17}+\frac{54351}{27647386}a^{14}+\frac{1}{234}a^{13}-\frac{1}{234}a^{12}+\frac{30108161}{248826474}a^{11}+\frac{17}{117}a^{10}-\frac{17}{117}a^{9}+\frac{369871}{248826474}a^{8}+\frac{19}{78}a^{7}-\frac{19}{78}a^{6}+\frac{180359}{490782}a^{5}-\frac{22}{117}a^{4}+\frac{22}{117}a^{3}+\frac{581093}{1472346}a^{2}+\frac{1}{18}a-\frac{1}{18}$
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
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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{52}{245391} a^{15} + \frac{2599}{245391} a^{12} - \frac{29792}{245391} a^{9} - \frac{92794}{245391} a^{6} + \frac{365716}{245391} a^{3} - \frac{58574}{245391} \)
(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: |
$\frac{103559}{248826474}a^{15}-\frac{5954603}{248826474}a^{12}+\frac{100890509}{248826474}a^{9}-\frac{411376399}{248826474}a^{6}+\frac{17591881}{19140498}a^{3}+\frac{338651}{1472346}$, $\frac{47555}{19140498}a^{15}-\frac{1332349}{9570249}a^{12}+\frac{42793079}{19140498}a^{9}-\frac{72718778}{9570249}a^{6}+\frac{157800031}{19140498}a^{3}-\frac{404408}{736173}$, $\frac{29674}{13823693}a^{17}+\frac{447652}{124413237}a^{15}-\frac{1700206}{13823693}a^{14}-\frac{25129921}{124413237}a^{12}+\frac{28748227}{13823693}a^{11}+\frac{405240028}{124413237}a^{9}-\frac{121683342}{13823693}a^{8}-\frac{1405021577}{124413237}a^{6}+\frac{13393286}{1063361}a^{5}+\frac{123943640}{9570249}a^{3}-\frac{435482}{81797}a^{2}-\frac{3621155}{736173}$, $\frac{47555}{19140498}a^{17}-\frac{139208}{124413237}a^{15}-\frac{1332349}{9570249}a^{14}+\frac{8109173}{124413237}a^{12}+\frac{42793079}{19140498}a^{11}-\frac{142437146}{124413237}a^{9}-\frac{72718778}{9570249}a^{8}+\frac{691185463}{124413237}a^{6}+\frac{157800031}{19140498}a^{5}-\frac{88609660}{9570249}a^{3}-\frac{404408}{736173}a^{2}+\frac{2393731}{736173}$, $\frac{704695}{248826474}a^{16}+\frac{163507}{124413237}a^{15}-\frac{40114861}{248826474}a^{13}-\frac{9347251}{124413237}a^{12}+\frac{668538085}{248826474}a^{10}+\frac{157843534}{124413237}a^{9}-\frac{2675966807}{248826474}a^{7}-\frac{683773679}{124413237}a^{6}+\frac{274935539}{19140498}a^{4}+\frac{102947693}{9570249}a^{3}-\frac{5026337}{1472346}a-\frac{6466508}{736173}$, $\frac{14276}{124413237}a^{17}-\frac{156119}{124413237}a^{16}+\frac{1065658}{124413237}a^{15}-\frac{921164}{124413237}a^{14}+\frac{7826543}{124413237}a^{13}-\frac{58859209}{124413237}a^{12}+\frac{19970402}{124413237}a^{11}-\frac{90929108}{124413237}a^{10}+\frac{913000198}{124413237}a^{9}-\frac{168383368}{124413237}a^{8}-\frac{242644574}{124413237}a^{7}-\frac{2594632592}{124413237}a^{6}+\frac{44790358}{9570249}a^{5}+\frac{56976587}{9570249}a^{4}+\frac{194742935}{9570249}a^{3}+\frac{649898}{736173}a^{2}-\frac{1222871}{736173}a-\frac{8254322}{736173}$, $\frac{20647}{9570249}a^{17}+\frac{4022887}{248826474}a^{16}-\frac{3468956}{124413237}a^{15}-\frac{79744}{736173}a^{14}-\frac{222269713}{248826474}a^{13}+\frac{377549311}{248826474}a^{12}+\frac{12175207}{9570249}a^{11}+\frac{3463283125}{248826474}a^{10}-\frac{2835415568}{124413237}a^{9}+\frac{24943108}{9570249}a^{8}-\frac{10107359057}{248826474}a^{7}+\frac{13448416523}{248826474}a^{6}+\frac{1778858}{9570249}a^{5}+\frac{598499561}{19140498}a^{4}-\frac{403713367}{9570249}a^{3}+\frac{579247}{736173}a^{2}-\frac{27890669}{1472346}a+\frac{37047635}{1472346}$, $\frac{152356}{9570249}a^{17}+\frac{4022887}{248826474}a^{16}+\frac{5789747}{248826474}a^{15}-\frac{648196}{736173}a^{14}-\frac{222269713}{248826474}a^{13}-\frac{321055043}{248826474}a^{12}+\frac{130860901}{9570249}a^{11}+\frac{3463283125}{248826474}a^{10}+\frac{5005136939}{248826474}a^{9}-\frac{368345837}{9570249}a^{8}-\frac{10107359057}{248826474}a^{7}-\frac{14443239925}{248826474}a^{6}+\frac{275431451}{9570249}a^{5}+\frac{598499561}{19140498}a^{4}+\frac{870564865}{19140498}a^{3}-\frac{12063293}{736173}a^{2}-\frac{27890669}{1472346}a-\frac{40891231}{1472346}$
| 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: | \( 81283585.10862277 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 81283585.10862277 \cdot 1}{6\cdot\sqrt{569752789846542015713684226048}}\cr\approx \mathstrut & 0.273922171002684 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 57*x^15 + 954*x^12 - 3931*x^9 + 6396*x^6 - 4563*x^3 + 2197)
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 - 57*x^15 + 954*x^12 - 3931*x^9 + 6396*x^6 - 4563*x^3 + 2197, 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 - 57*x^15 + 954*x^12 - 3931*x^9 + 6396*x^6 - 4563*x^3 + 2197);
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 - 57*x^15 + 954*x^12 - 3931*x^9 + 6396*x^6 - 4563*x^3 + 2197);
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
$C_3\times S_3^2$ (as 18T46):
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 108 |
| The 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.216.1, 6.0.139968.1, 6.0.29937843.3 |
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 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.234952967714936979456.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
|
\(3\)
| Deg $18$ | $18$ | $1$ | $37$ | |||
|
\(13\)
| 13.3.2.3 | $x^{3} + 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.6.4.2 | $x^{6} - 156 x^{3} + 338$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |