Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 57024*x^8 + 88704*x^6 - 69120*x^4 + 20736*x^2 - 512)
gp: K = bnfinit(y^18 - 36*y^16 + 540*y^14 - 4368*y^12 + 20592*y^10 - 57024*y^8 + 88704*y^6 - 69120*y^4 + 20736*y^2 - 512, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 57024*x^8 + 88704*x^6 - 69120*x^4 + 20736*x^2 - 512);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 57024*x^8 + 88704*x^6 - 69120*x^4 + 20736*x^2 - 512)
\( x^{18} - 36 x^{16} + 540 x^{14} - 4368 x^{12} + 20592 x^{10} - 57024 x^{8} + 88704 x^{6} - 69120 x^{4} + \cdots - 512 \)
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: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(132173713091594538512566714368\)
\(\medspace = 2^{27}\cdot 3^{44}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(41.48\) | 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: | $2^{3/2}3^{22/9}\approx 41.48025274304532$ | ||
| 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: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(216=2^{3}\cdot 3^{3}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{216}(1,·)$, $\chi_{216}(133,·)$, $\chi_{216}(193,·)$, $\chi_{216}(73,·)$, $\chi_{216}(13,·)$, $\chi_{216}(205,·)$, $\chi_{216}(145,·)$, $\chi_{216}(85,·)$, $\chi_{216}(25,·)$, $\chi_{216}(157,·)$, $\chi_{216}(97,·)$, $\chi_{216}(37,·)$, $\chi_{216}(169,·)$, $\chi_{216}(109,·)$, $\chi_{216}(49,·)$, $\chi_{216}(181,·)$, $\chi_{216}(121,·)$, $\chi_{216}(61,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$
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: | $17$ | 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: |
$\frac{1}{64}a^{12}-\frac{3}{8}a^{10}+\frac{27}{8}a^{8}-\frac{111}{8}a^{6}+\frac{99}{4}a^{4}-\frac{27}{2}a^{2}+1$, $\frac{1}{8}a^{6}-\frac{3}{2}a^{4}+\frac{9}{2}a^{2}-3$, $\frac{1}{4}a^{4}-2a^{2}+2$, $\frac{1}{2}a^{2}-2$, $\frac{1}{128}a^{14}-\frac{13}{64}a^{12}+\frac{33}{16}a^{10}-\frac{165}{16}a^{8}+\frac{209}{8}a^{6}-\frac{121}{4}a^{4}+\frac{21}{2}a^{2}+1$, $\frac{1}{32}a^{10}-\frac{5}{8}a^{8}+\frac{35}{8}a^{6}-\frac{25}{2}a^{4}+\frac{25}{2}a^{2}-2$, $\frac{1}{16}a^{8}-a^{6}+5a^{4}-8a^{2}+2$, $\frac{1}{128}a^{14}-\frac{7}{32}a^{12}+\frac{77}{32}a^{10}-\frac{105}{8}a^{8}+\frac{147}{4}a^{6}-\frac{195}{4}a^{4}+\frac{45}{2}a^{2}$, $\frac{1}{16}a^{9}-\frac{9}{8}a^{7}+\frac{27}{4}a^{5}-15a^{3}+9a-1$, $\frac{1}{64}a^{12}-\frac{3}{8}a^{10}-\frac{1}{16}a^{9}+\frac{27}{8}a^{8}+\frac{9}{8}a^{7}-\frac{111}{8}a^{6}-\frac{27}{4}a^{5}+\frac{99}{4}a^{4}+15a^{3}-\frac{27}{2}a^{2}-9a$, $\frac{1}{32}a^{10}-\frac{1}{16}a^{9}-\frac{5}{8}a^{8}+\frac{9}{8}a^{7}+\frac{35}{8}a^{6}-\frac{27}{4}a^{5}-\frac{25}{2}a^{4}+15a^{3}+\frac{25}{2}a^{2}-9a-2$, $\frac{1}{32}a^{10}-\frac{1}{16}a^{9}-\frac{9}{16}a^{8}+\frac{9}{8}a^{7}+\frac{27}{8}a^{6}-\frac{27}{4}a^{5}-\frac{15}{2}a^{4}+15a^{3}+\frac{9}{2}a^{2}-9a$, $\frac{1}{16}a^{9}-\frac{9}{8}a^{7}-\frac{1}{8}a^{6}+\frac{27}{4}a^{5}+\frac{3}{2}a^{4}-15a^{3}-\frac{9}{2}a^{2}+9a+2$, $\frac{1}{16}a^{9}-\frac{9}{8}a^{7}+\frac{27}{4}a^{5}+\frac{1}{4}a^{4}-15a^{3}-2a^{2}+9a+2$, $\frac{1}{256}a^{16}-\frac{1}{8}a^{14}+\frac{13}{8}a^{12}-11a^{10}+\frac{1}{16}a^{9}+\frac{165}{4}a^{8}-\frac{9}{8}a^{7}-84a^{6}+\frac{27}{4}a^{5}+84a^{4}-15a^{3}-32a^{2}+9a+2$, $\frac{1}{128}a^{14}+\frac{1}{64}a^{13}-\frac{13}{64}a^{12}-\frac{3}{8}a^{11}+\frac{33}{16}a^{10}+\frac{55}{16}a^{9}-\frac{165}{16}a^{8}-15a^{7}+\frac{211}{8}a^{6}+\frac{63}{2}a^{5}-\frac{131}{4}a^{4}-28a^{3}+\frac{33}{2}a^{2}+7a-1$, $\frac{1}{128}a^{14}-\frac{7}{32}a^{12}+\frac{77}{32}a^{10}-\frac{1}{16}a^{9}-\frac{105}{8}a^{8}+\frac{9}{8}a^{7}+\frac{147}{4}a^{6}-\frac{27}{4}a^{5}-\frac{195}{4}a^{4}+15a^{3}+\frac{45}{2}a^{2}-9a$
| 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: | \( 336692032.456 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 336692032.456 \cdot 1}{2\cdot\sqrt{132173713091594538512566714368}}\cr\approx \mathstrut & 0.121386452739 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 57024*x^8 + 88704*x^6 - 69120*x^4 + 20736*x^2 - 512)
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 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 57024*x^8 + 88704*x^6 - 69120*x^4 + 20736*x^2 - 512, 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 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 57024*x^8 + 88704*x^6 - 69120*x^4 + 20736*x^2 - 512);
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 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 57024*x^8 + 88704*x^6 - 69120*x^4 + 20736*x^2 - 512);
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 cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 6.6.3359232.1, \(\Q(\zeta_{27})^+\) |
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]
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 | $18$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.27.119 | $x^{18} + 70 x^{16} + 1024 x^{15} - 9632 x^{14} + 37696 x^{13} + 414592 x^{12} - 7305728 x^{11} + 60591136 x^{10} - 292750080 x^{9} + 723623360 x^{8} - 693690368 x^{7} + 2330844160 x^{6} - 20528915456 x^{5} + 72224523264 x^{4} - 130591481856 x^{3} + 133128467712 x^{2} - 74944811008 x + 19148434944$ | $2$ | $9$ | $27$ | $C_{18}$ | $[3]^{9}$ |
|
\(3\)
| Deg $18$ | $9$ | $2$ | $44$ |