Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^15 + 24*x^12 - 74*x^9 + 150*x^6 + 12*x^3 + 1)
gp: K = bnfinit(y^18 - 6*y^15 + 24*y^12 - 74*y^9 + 150*y^6 + 12*y^3 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^15 + 24*x^12 - 74*x^9 + 150*x^6 + 12*x^3 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^15 + 24*x^12 - 74*x^9 + 150*x^6 + 12*x^3 + 1)
\( x^{18} - 6x^{15} + 24x^{12} - 74x^{9} + 150x^{6} + 12x^{3} + 1 \)
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: |
\(-2954312706550833698643\)
\(\medspace = -\,3^{45}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.59\) | 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: | $3^{47/18}\approx 17.6123217011059$ | ||
| Ramified primes: |
\(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{6}a^{9}-\frac{1}{6}$, $\frac{1}{6}a^{10}-\frac{1}{6}a$, $\frac{1}{18}a^{11}+\frac{1}{18}a^{10}+\frac{1}{18}a^{9}-\frac{1}{18}a^{2}-\frac{1}{18}a-\frac{1}{18}$, $\frac{1}{18}a^{12}-\frac{1}{18}a^{9}-\frac{1}{18}a^{3}+\frac{1}{18}$, $\frac{1}{18}a^{13}-\frac{1}{18}a^{10}-\frac{1}{18}a^{4}+\frac{1}{18}a$, $\frac{1}{18}a^{14}+\frac{1}{18}a^{10}+\frac{1}{18}a^{9}-\frac{1}{18}a^{5}-\frac{1}{18}a-\frac{1}{18}$, $\frac{1}{306}a^{15}-\frac{2}{153}a^{12}-\frac{1}{17}a^{9}-\frac{25}{306}a^{6}-\frac{1}{153}a^{3}-\frac{10}{51}$, $\frac{1}{918}a^{16}-\frac{1}{918}a^{15}+\frac{13}{918}a^{13}-\frac{13}{918}a^{12}+\frac{8}{459}a^{10}-\frac{8}{459}a^{9}-\frac{127}{918}a^{7}+\frac{127}{918}a^{6}+\frac{185}{918}a^{4}-\frac{185}{918}a^{3}-\frac{98}{459}a+\frac{98}{459}$, $\frac{1}{918}a^{17}-\frac{1}{918}a^{15}+\frac{13}{918}a^{14}-\frac{13}{918}a^{12}+\frac{8}{459}a^{11}-\frac{8}{459}a^{9}-\frac{127}{918}a^{8}+\frac{127}{918}a^{6}+\frac{185}{918}a^{5}-\frac{185}{918}a^{3}-\frac{98}{459}a^{2}+\frac{98}{459}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
\( \frac{4}{51} a^{15} - \frac{49}{102} a^{12} + \frac{98}{51} a^{9} - \frac{304}{51} a^{6} + \frac{1225}{102} a^{3} + \frac{49}{51} \)
(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: |
$\frac{13}{306}a^{17}-\frac{43}{153}a^{14}+\frac{361}{306}a^{11}-\frac{1141}{306}a^{8}+\frac{1279}{153}a^{5}-\frac{967}{306}a^{2}$, $\frac{13}{306}a^{16}-\frac{43}{153}a^{13}+\frac{361}{306}a^{10}-\frac{1141}{306}a^{7}+\frac{1279}{153}a^{4}-\frac{967}{306}a$, $\frac{193}{918}a^{17}-\frac{41}{918}a^{16}+\frac{19}{918}a^{15}-\frac{1163}{918}a^{14}+\frac{116}{459}a^{13}-\frac{55}{459}a^{12}+\frac{4669}{918}a^{11}-\frac{911}{918}a^{10}+\frac{203}{459}a^{9}-\frac{14413}{918}a^{8}+\frac{2759}{918}a^{7}-\frac{1189}{918}a^{6}+\frac{29279}{918}a^{5}-\frac{2645}{459}a^{4}+\frac{1171}{459}a^{3}+\frac{1901}{918}a^{2}-\frac{2419}{918}a+\frac{535}{459}$, $\frac{2}{153}a^{17}+\frac{1}{18}a^{16}+\frac{1}{18}a^{15}-\frac{8}{153}a^{14}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{47}{306}a^{11}+\frac{4}{3}a^{10}+\frac{4}{3}a^{9}-\frac{50}{153}a^{8}-\frac{73}{18}a^{7}-\frac{73}{18}a^{6}-\frac{4}{153}a^{5}+\frac{25}{3}a^{4}+\frac{25}{3}a^{3}+\frac{1171}{306}a^{2}+\frac{2}{3}a+\frac{2}{3}$, $\frac{29}{918}a^{17}-\frac{73}{918}a^{16}+\frac{11}{918}a^{15}-\frac{92}{459}a^{14}+\frac{214}{459}a^{13}-\frac{61}{918}a^{12}+\frac{385}{459}a^{11}-\frac{1729}{918}a^{10}+\frac{139}{459}a^{9}-\frac{2459}{918}a^{8}+\frac{5293}{918}a^{7}-\frac{785}{918}a^{6}+\frac{2657}{459}a^{5}-\frac{5299}{459}a^{4}+\frac{1933}{918}a^{3}-\frac{1006}{459}a^{2}-\frac{1043}{918}a+\frac{95}{459}$, $\frac{161}{918}a^{17}-\frac{73}{918}a^{16}-\frac{1}{918}a^{15}-\frac{967}{918}a^{14}+\frac{214}{459}a^{13}-\frac{13}{918}a^{12}+\frac{3851}{918}a^{11}-\frac{1729}{918}a^{10}+\frac{35}{918}a^{9}-\frac{11879}{918}a^{8}+\frac{5293}{918}a^{7}-\frac{179}{918}a^{6}+\frac{23971}{918}a^{5}-\frac{5299}{459}a^{4}+\frac{427}{918}a^{3}+\frac{2359}{918}a^{2}-\frac{1043}{918}a-\frac{161}{918}$, $\frac{76}{459}a^{17}+\frac{13}{459}a^{16}-\frac{67}{918}a^{15}-\frac{931}{918}a^{14}-\frac{86}{459}a^{13}+\frac{202}{459}a^{12}+\frac{1879}{459}a^{11}+\frac{361}{459}a^{10}-\frac{1633}{918}a^{9}-\frac{5827}{459}a^{8}-\frac{1192}{459}a^{7}+\frac{5143}{918}a^{6}+\frac{23989}{918}a^{5}+\frac{2660}{459}a^{4}-\frac{5305}{459}a^{3}-\frac{259}{459}a^{2}-\frac{1324}{459}a-\frac{689}{918}$, $\frac{41}{918}a^{17}+\frac{13}{459}a^{16}+\frac{7}{459}a^{15}-\frac{116}{459}a^{14}-\frac{86}{459}a^{13}-\frac{73}{918}a^{12}+\frac{911}{918}a^{11}+\frac{361}{459}a^{10}+\frac{275}{918}a^{9}-\frac{2759}{918}a^{8}-\frac{1192}{459}a^{7}-\frac{430}{459}a^{6}+\frac{2645}{459}a^{5}+\frac{2660}{459}a^{4}+\frac{1927}{918}a^{3}+\frac{2419}{918}a^{2}-\frac{1324}{459}a+\frac{877}{918}$
| 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: | \( 12334.7656984 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 12334.7656984 \cdot 1}{6\cdot\sqrt{2954312706550833698643}}\cr\approx \mathstrut & 0.577258369085 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^15 + 24*x^12 - 74*x^9 + 150*x^6 + 12*x^3 + 1)
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 - 6*x^15 + 24*x^12 - 74*x^9 + 150*x^6 + 12*x^3 + 1, 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 - 6*x^15 + 24*x^12 - 74*x^9 + 150*x^6 + 12*x^3 + 1);
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 - 6*x^15 + 24*x^12 - 74*x^9 + 150*x^6 + 12*x^3 + 1);
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
$C_3^2:C_6$ (as 18T21):
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 54 |
| The 10 conjugacy class representatives for $C_3^2:C_6$ |
| Character table for $C_3^2:C_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 6.0.177147.2, 9.1.31381059609.1 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 9 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Minimal sibling: | 9.1.31381059609.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $18$ | $1$ | $45$ |