Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 105*x^14 - 63*x^13 - 21*x^12 + 72*x^11 - 63*x^10 + 27*x^9 - 6*x^6 - 9*x^5 + 18*x^4 - 9*x^2 + 3)
gp: K = bnfinit(y^18 - 9*y^17 + 36*y^16 - 81*y^15 + 105*y^14 - 63*y^13 - 21*y^12 + 72*y^11 - 63*y^10 + 27*y^9 - 6*y^6 - 9*y^5 + 18*y^4 - 9*y^2 + 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 105*x^14 - 63*x^13 - 21*x^12 + 72*x^11 - 63*x^10 + 27*x^9 - 6*x^6 - 9*x^5 + 18*x^4 - 9*x^2 + 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 105*x^14 - 63*x^13 - 21*x^12 + 72*x^11 - 63*x^10 + 27*x^9 - 6*x^6 - 9*x^5 + 18*x^4 - 9*x^2 + 3)
\( x^{18} - 9 x^{17} + 36 x^{16} - 81 x^{15} + 105 x^{14} - 63 x^{13} - 21 x^{12} + 72 x^{11} - 63 x^{10} + \cdots + 3 \)
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: |
\(-2529990231179046912\)
\(\medspace = -\,2^{12}\cdot 3^{31}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.53\) | 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: | $2^{2/3}3^{31/18}\approx 10.529202818387967$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{159031}a^{17}+\frac{66377}{159031}a^{16}+\frac{72610}{159031}a^{15}+\frac{57769}{159031}a^{14}+\frac{20374}{159031}a^{13}-\frac{10354}{159031}a^{12}-\frac{28683}{159031}a^{11}-\frac{71403}{159031}a^{10}+\frac{77396}{159031}a^{9}+\frac{37335}{159031}a^{8}+\frac{23175}{159031}a^{7}+\frac{29656}{159031}a^{6}-\frac{60570}{159031}a^{5}-\frac{60225}{159031}a^{4}-\frac{57492}{159031}a^{3}-\frac{78943}{159031}a^{2}-\frac{2433}{159031}a+\frac{58358}{159031}$
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 $1$
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: |
\( -\frac{160828}{159031} a^{17} + \frac{1425060}{159031} a^{16} - \frac{5322773}{159031} a^{15} + \frac{10373365}{159031} a^{14} - \frac{9576808}{159031} a^{13} - \frac{636613}{159031} a^{12} + \frac{10672384}{159031} a^{11} - \frac{10840934}{159031} a^{10} + \frac{5478567}{159031} a^{9} - \frac{457006}{159031} a^{8} - \frac{1569663}{159031} a^{7} - \frac{652571}{159031} a^{6} + \frac{862241}{159031} a^{5} + \frac{3263865}{159031} a^{4} - \frac{1806367}{159031} a^{3} - \frac{1744422}{159031} a^{2} + \frac{555357}{159031} a + \frac{886289}{159031} \)
(order $18$)
| 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: |
$\frac{153946}{159031}a^{17}-\frac{1176480}{159031}a^{16}+\frac{3864876}{159031}a^{15}-\frac{6545379}{159031}a^{14}+\frac{4698321}{159031}a^{13}+\frac{2237263}{159031}a^{12}-\frac{7134767}{159031}a^{11}+\frac{5582567}{159031}a^{10}-\frac{1547245}{159031}a^{9}-\frac{760616}{159031}a^{8}+\frac{792251}{159031}a^{7}+\frac{914814}{159031}a^{6}-\frac{362659}{159031}a^{5}-\frac{1957953}{159031}a^{4}+\frac{842997}{159031}a^{3}+\frac{1462190}{159031}a^{2}-\frac{191644}{159031}a-\frac{793739}{159031}$, $\frac{205995}{159031}a^{17}-\frac{1745575}{159031}a^{16}+\frac{6474578}{159031}a^{15}-\frac{13205117}{159031}a^{14}+\frac{14903923}{159031}a^{13}-\frac{6945822}{159031}a^{12}-\frac{3256462}{159031}a^{11}+\frac{6636476}{159031}a^{10}-\frac{4598691}{159031}a^{9}+\frac{1356413}{159031}a^{8}+\frac{300598}{159031}a^{7}+\frac{1561196}{159031}a^{6}-\frac{1453262}{159031}a^{5}-\frac{2266999}{159031}a^{4}+\frac{1405309}{159031}a^{3}+\frac{1600961}{159031}a^{2}-\frac{556247}{159031}a-\frac{492235}{159031}$, $\frac{266986}{159031}a^{17}-\frac{2275228}{159031}a^{16}+\frac{8403203}{159031}a^{15}-\frac{16805556}{159031}a^{14}+\frac{17887912}{159031}a^{13}-\frac{6139380}{159031}a^{12}-\frac{7137059}{159031}a^{11}+\frac{9940658}{159031}a^{10}-\frac{6087707}{159031}a^{9}+\frac{1449540}{159031}a^{8}+\frac{935619}{159031}a^{7}+\frac{1650729}{159031}a^{6}-\frac{1388002}{159031}a^{5}-\frac{3583215}{159031}a^{4}+\frac{1862349}{159031}a^{3}+\frac{2406159}{159031}a^{2}-\frac{571427}{159031}a-\frac{770330}{159031}$, $\frac{556695}{159031}a^{17}-\frac{4038796}{159031}a^{16}+\frac{12323943}{159031}a^{15}-\frac{18460054}{159031}a^{14}+\frac{9077777}{159031}a^{13}+\frac{11985890}{159031}a^{12}-\frac{21326253}{159031}a^{11}+\frac{13782031}{159031}a^{10}-\frac{3165168}{159031}a^{9}-\frac{3847402}{159031}a^{8}+\frac{3038339}{159031}a^{7}+\frac{2883306}{159031}a^{6}+\frac{1917090}{159031}a^{5}-\frac{6402195}{159031}a^{4}-\frac{679221}{159031}a^{3}+\frac{3746992}{159031}a^{2}+\frac{823247}{159031}a-\frac{836180}{159031}$, $\frac{471110}{159031}a^{17}-\frac{3531866}{159031}a^{16}+\frac{11338263}{159031}a^{15}-\frac{18823222}{159031}a^{14}+\frac{13914832}{159031}a^{13}+\frac{3424574}{159031}a^{12}-\frac{15251036}{159031}a^{11}+\frac{13882580}{159031}a^{10}-\frac{6959360}{159031}a^{9}-\frac{254812}{159031}a^{8}+\frac{2086410}{159031}a^{7}+\frac{1796089}{159031}a^{6}+\frac{1389940}{159031}a^{5}-\frac{5127063}{159031}a^{4}+\frac{149614}{159031}a^{3}+\frac{2538395}{159031}a^{2}+\frac{402880}{159031}a-\frac{400931}{159031}$, $\frac{178216}{159031}a^{17}-\frac{1190720}{159031}a^{16}+\frac{3409972}{159031}a^{15}-\frac{5236797}{159031}a^{14}+\frac{4906953}{159031}a^{13}-\frac{4623670}{159031}a^{12}+\frac{6325145}{159031}a^{11}-\frac{5857668}{159031}a^{10}+\frac{2832371}{159031}a^{9}-\frac{480742}{159031}a^{8}-\frac{1946673}{159031}a^{7}+\frac{2959031}{159031}a^{6}-\frac{154964}{159031}a^{5}+\frac{579714}{159031}a^{4}-\frac{2012407}{159031}a^{3}+\frac{89789}{159031}a^{2}+\frac{873164}{159031}a+\frac{497083}{159031}$, $\frac{70412}{159031}a^{17}-\frac{978921}{159031}a^{16}+\frac{5016693}{159031}a^{15}-\frac{13422694}{159031}a^{14}+\frac{20152374}{159031}a^{13}-\frac{15314720}{159031}a^{12}+\frac{1656614}{159031}a^{11}+\frac{6657300}{159031}a^{10}-\frac{6905489}{159031}a^{9}+\frac{4184396}{159031}a^{8}-\frac{814146}{159031}a^{7}+\frac{1015428}{159031}a^{6}-\frac{3301133}{159031}a^{5}-\frac{1273333}{159031}a^{4}+\frac{3347052}{159031}a^{3}+\frac{871182}{159031}a^{2}-\frac{1626319}{159031}a-\frac{893668}{159031}$, $\frac{600349}{159031}a^{17}-\frac{4439151}{159031}a^{16}+\frac{13825301}{159031}a^{15}-\frac{21229253}{159031}a^{14}+\frac{10773331}{159031}a^{13}+\frac{15139096}{159031}a^{12}-\frac{29513453}{159031}a^{11}+\frac{22509774}{159031}a^{10}-\frac{8540833}{159031}a^{9}-\frac{2661783}{159031}a^{8}+\frac{4395846}{159031}a^{7}+\frac{1542711}{159031}a^{6}+\frac{2797902}{159031}a^{5}-\frac{8272225}{159031}a^{4}+\frac{664501}{159031}a^{3}+\frac{3930071}{159031}a^{2}+\frac{527711}{159031}a-\frac{1111699}{159031}$
| 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: | \( 404.056392969 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 404.056392969 \cdot 1}{18\cdot\sqrt{2529990231179046912}}\cr\approx \mathstrut & 0.215391653651 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 105*x^14 - 63*x^13 - 21*x^12 + 72*x^11 - 63*x^10 + 27*x^9 - 6*x^6 - 9*x^5 + 18*x^4 - 9*x^2 + 3)
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 - 9*x^17 + 36*x^16 - 81*x^15 + 105*x^14 - 63*x^13 - 21*x^12 + 72*x^11 - 63*x^10 + 27*x^9 - 6*x^6 - 9*x^5 + 18*x^4 - 9*x^2 + 3, 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 - 9*x^17 + 36*x^16 - 81*x^15 + 105*x^14 - 63*x^13 - 21*x^12 + 72*x^11 - 63*x^10 + 27*x^9 - 6*x^6 - 9*x^5 + 18*x^4 - 9*x^2 + 3);
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 - 9*x^17 + 36*x^16 - 81*x^15 + 105*x^14 - 63*x^13 - 21*x^12 + 72*x^11 - 63*x^10 + 27*x^9 - 6*x^6 - 9*x^5 + 18*x^4 - 9*x^2 + 3);
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
$C_3\times S_3$ (as 18T3):
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 18 |
| The 9 conjugacy class representatives for $S_3 \times C_3$ |
| Character table for $S_3 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.108.1 x3, \(\Q(\zeta_{9})^+\), 6.0.34992.1, 6.0.314928.1 x2, \(\Q(\zeta_{9})\), 9.3.918330048.1 x3 |
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 6 sibling: | 6.0.314928.1 |
| Degree 9 sibling: | 9.3.918330048.1 |
| Minimal sibling: | 6.0.314928.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| Deg $18$ | $18$ | $1$ | $31$ |