Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12691*x^9 + 40353607)
gp: K = bnfinit(y^18 - 12691*y^9 + 40353607, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 12691*x^9 + 40353607);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 12691*x^9 + 40353607)
\( x^{18} - 12691x^{9} + 40353607 \)
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: |
\(-40891490652933723328906846968243\)
\(\medspace = -\,3^{45}\cdot 7^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(57.04\) | 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: | $3^{5/2}7^{2/3}\approx 57.042930691680226$ | ||
| Ramified primes: |
\(3\), \(7\)
| 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 and abelian over $\Q$. | |||
| Conductor: | \(189=3^{3}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{189}(128,·)$, $\chi_{189}(1,·)$, $\chi_{189}(2,·)$, $\chi_{189}(67,·)$, $\chi_{189}(4,·)$, $\chi_{189}(134,·)$, $\chi_{189}(71,·)$, $\chi_{189}(8,·)$, $\chi_{189}(130,·)$, $\chi_{189}(142,·)$, $\chi_{189}(79,·)$, $\chi_{189}(16,·)$, $\chi_{189}(158,·)$, $\chi_{189}(95,·)$, $\chi_{189}(32,·)$, $\chi_{189}(65,·)$, $\chi_{189}(64,·)$, $\chi_{189}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{7}a^{3}$, $\frac{1}{7}a^{4}$, $\frac{1}{7}a^{5}$, $\frac{1}{49}a^{6}$, $\frac{1}{49}a^{7}$, $\frac{1}{49}a^{8}$, $\frac{1}{343}a^{9}$, $\frac{1}{343}a^{10}$, $\frac{1}{2401}a^{11}-\frac{2}{7}a^{2}$, $\frac{1}{16807}a^{12}-\frac{2}{49}a^{3}$, $\frac{1}{16807}a^{13}-\frac{2}{49}a^{4}$, $\frac{1}{117649}a^{14}+\frac{12}{343}a^{5}$, $\frac{1}{823543}a^{15}+\frac{12}{2401}a^{6}$, $\frac{1}{823543}a^{16}+\frac{12}{2401}a^{7}$, $\frac{1}{5764801}a^{17}-\frac{37}{16807}a^{8}$
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
$C_{543}$, which has order $543$ (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: |
\( -\frac{3}{823543} a^{15} + \frac{62}{2401} a^{6} \)
(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: |
$\frac{5}{823543}a^{15}+\frac{3}{16807}a^{12}-\frac{87}{2401}a^{6}-\frac{55}{49}a^{3}$, $\frac{8}{823543}a^{15}+\frac{1}{16807}a^{12}-\frac{149}{2401}a^{6}-\frac{16}{49}a^{3}$, $\frac{1}{5764801}a^{17}-\frac{5}{823543}a^{16}-\frac{32}{823543}a^{15}-\frac{3}{117649}a^{14}+\frac{1}{16807}a^{13}+\frac{2}{16807}a^{12}+\frac{3}{2401}a^{11}+\frac{1}{343}a^{10}-\frac{37}{16807}a^{8}+\frac{87}{2401}a^{7}+\frac{596}{2401}a^{6}+\frac{62}{343}a^{5}-\frac{16}{49}a^{4}-\frac{32}{49}a^{3}-\frac{55}{7}a^{2}-19a-2$, $\frac{19}{5764801}a^{17}+\frac{5}{823543}a^{16}-\frac{16}{823543}a^{15}+\frac{3}{117649}a^{14}+\frac{3}{16807}a^{13}+\frac{1}{16807}a^{12}-\frac{360}{16807}a^{8}-\frac{87}{2401}a^{7}+\frac{298}{2401}a^{6}-\frac{62}{343}a^{5}-\frac{55}{49}a^{4}-\frac{16}{49}a^{3}-1$, $\frac{18}{5764801}a^{17}+\frac{8}{823543}a^{16}+\frac{5}{823543}a^{15}-\frac{3}{16807}a^{13}-\frac{6}{16807}a^{12}-\frac{3}{2401}a^{11}+\frac{1}{343}a^{9}-\frac{323}{16807}a^{8}-\frac{149}{2401}a^{7}-\frac{87}{2401}a^{6}+\frac{55}{49}a^{4}+\frac{110}{49}a^{3}+\frac{55}{7}a^{2}-19$, $\frac{1}{5764801}a^{17}+\frac{8}{823543}a^{16}-\frac{13}{823543}a^{15}-\frac{5}{117649}a^{14}-\frac{8}{16807}a^{12}+\frac{1}{343}a^{10}+\frac{1}{343}a^{9}-\frac{37}{16807}a^{8}-\frac{149}{2401}a^{7}+\frac{236}{2401}a^{6}+\frac{87}{343}a^{5}+\frac{156}{49}a^{3}-18a-17$, $\frac{1}{5764801}a^{17}-\frac{3}{823543}a^{16}+\frac{5}{823543}a^{15}+\frac{2}{16807}a^{13}-\frac{6}{16807}a^{12}+\frac{1}{2401}a^{11}+\frac{1}{343}a^{9}-\frac{37}{16807}a^{8}+\frac{62}{2401}a^{7}-\frac{87}{2401}a^{6}-\frac{39}{49}a^{4}+\frac{110}{49}a^{3}-\frac{16}{7}a^{2}-19$, $\frac{1}{5764801}a^{17}-\frac{3}{823543}a^{16}-\frac{2}{823543}a^{15}+\frac{1}{16807}a^{12}+\frac{1}{2401}a^{11}-\frac{37}{16807}a^{8}+\frac{62}{2401}a^{7}+\frac{25}{2401}a^{6}-\frac{23}{49}a^{3}-\frac{16}{7}a^{2}-a+4$
| 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: | \( 4392158.29124 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 4392158.29124 \cdot 543}{18\cdot\sqrt{40891490652933723328906846968243}}\cr\approx \mathstrut & 0.316233337742 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 12691*x^9 + 40353607)
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 - 12691*x^9 + 40353607, 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 - 12691*x^9 + 40353607);
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 - 12691*x^9 + 40353607);
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
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 9.9.3691950281939241.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18$ | R | $18$ | R | $18$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | $18$ | $18$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{18}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | $18$ |
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 $18$ | $18$ | $1$ | $45$ | |||
|
\(7\)
| 7.9.6.2 | $x^{9} + 252 x^{6} - 2352 x^{3} + 5488$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |
| 7.9.6.2 | $x^{9} + 252 x^{6} - 2352 x^{3} + 5488$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |