Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 171*x^12 + 9747*x^6 + 27)
gp: K = bnfinit(y^18 - 171*y^12 + 9747*y^6 + 27, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 171*x^12 + 9747*x^6 + 27);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 171*x^12 + 9747*x^6 + 27)
\( x^{18} - 171x^{12} + 9747x^{6} + 27 \)
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: |
\(-450283905890997363000000000000\)
\(\medspace = -\,2^{12}\cdot 3^{37}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(44.40\) | 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: | $2^{2/3}3^{37/18}5^{2/3}\approx 44.40336798307826$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{21}a^{6}+\frac{2}{7}$, $\frac{1}{21}a^{7}+\frac{2}{7}a$, $\frac{1}{21}a^{8}+\frac{2}{7}a^{2}$, $\frac{1}{42}a^{9}-\frac{1}{42}a^{6}-\frac{5}{14}a^{3}+\frac{5}{14}$, $\frac{1}{126}a^{10}-\frac{1}{42}a^{7}-\frac{5}{42}a^{4}+\frac{5}{14}a$, $\frac{1}{126}a^{11}-\frac{1}{42}a^{8}-\frac{5}{42}a^{5}+\frac{5}{14}a^{2}$, $\frac{1}{1764}a^{12}+\frac{1}{147}a^{6}-\frac{1}{2}a^{3}+\frac{53}{196}$, $\frac{1}{5292}a^{13}-\frac{1}{126}a^{9}+\frac{8}{441}a^{7}-\frac{1}{42}a^{6}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{5}{42}a^{3}+\frac{109}{588}a+\frac{5}{14}$, $\frac{1}{5292}a^{14}+\frac{8}{441}a^{8}-\frac{1}{2}a^{5}+\frac{109}{588}a^{2}$, $\frac{1}{5292}a^{15}-\frac{5}{882}a^{9}-\frac{269}{588}a^{3}-\frac{1}{2}$, $\frac{1}{5292}a^{16}+\frac{1}{441}a^{10}-\frac{1}{42}a^{7}+\frac{83}{196}a^{4}-\frac{1}{7}a$, $\frac{1}{5292}a^{17}+\frac{1}{441}a^{11}-\frac{1}{42}a^{8}+\frac{83}{196}a^{5}-\frac{1}{7}a^{2}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $7$ |
Class group and class number
$C_{3}\times C_{6}\times C_{6}$, which has order $108$ (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: |
\( -\frac{1}{588} a^{15} + \frac{85}{294} a^{9} - \frac{3225}{196} a^{3} + \frac{1}{2} \)
(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: |
$\frac{1}{5292}a^{15}-\frac{1}{1764}a^{12}-\frac{13}{441}a^{9}+\frac{19}{294}a^{6}+\frac{823}{588}a^{3}+\frac{31}{196}$, $\frac{1}{2646}a^{15}+\frac{1}{1764}a^{12}-\frac{26}{441}a^{9}-\frac{2}{49}a^{6}+\frac{485}{147}a^{3}-\frac{3}{196}$, $\frac{25}{5292}a^{17}-\frac{1}{2646}a^{16}+\frac{1}{1764}a^{14}-\frac{713}{882}a^{11}+\frac{59}{882}a^{10}-\frac{13}{147}a^{8}-\frac{1}{42}a^{7}+\frac{27085}{588}a^{5}-\frac{192}{49}a^{4}+\frac{1019}{196}a^{2}+\frac{33}{14}a$, $\frac{25}{2646}a^{17}+\frac{1}{294}a^{16}+\frac{1}{882}a^{15}+\frac{1}{2646}a^{14}-\frac{713}{441}a^{11}-\frac{85}{147}a^{10}-\frac{59}{294}a^{9}-\frac{26}{441}a^{8}+\frac{1}{42}a^{6}+\frac{27085}{294}a^{5}+\frac{3225}{98}a^{4}+\frac{576}{49}a^{3}+\frac{823}{294}a^{2}-\frac{5}{14}$, $\frac{65}{5292}a^{17}-\frac{13}{2646}a^{16}+\frac{1}{5292}a^{15}+\frac{1}{1764}a^{13}-\frac{1}{882}a^{12}-\frac{617}{294}a^{11}+\frac{739}{882}a^{10}-\frac{13}{441}a^{9}-\frac{11}{98}a^{7}+\frac{73}{294}a^{6}+\frac{70099}{588}a^{5}-\frac{13807}{294}a^{4}-\frac{59}{588}a^{3}+\frac{7}{2}a^{2}-\frac{185}{196}a+\frac{3}{98}$, $\frac{205}{588}a^{17}-\frac{1}{6}a^{16}+\frac{319}{5292}a^{15}-\frac{1}{63}a^{14}+\frac{1}{588}a^{13}+\frac{1}{882}a^{12}-\frac{52583}{882}a^{11}+\frac{57}{2}a^{10}-\frac{4546}{441}a^{9}+\frac{19}{7}a^{8}-\frac{85}{294}a^{7}-\frac{59}{294}a^{6}+\frac{1998215}{588}a^{5}-\frac{3249}{2}a^{4}+\frac{345487}{588}a^{3}-\frac{2165}{14}a^{2}+\frac{3225}{196}a+\frac{1103}{98}$, $\frac{11}{5292}a^{17}-\frac{13}{1764}a^{16}-\frac{1}{1323}a^{15}+\frac{1}{882}a^{14}+\frac{1}{1764}a^{12}-\frac{157}{441}a^{11}+\frac{556}{441}a^{10}+\frac{52}{441}a^{9}-\frac{59}{294}a^{8}-\frac{1}{42}a^{7}-\frac{2}{49}a^{6}+\frac{3993}{196}a^{5}-\frac{42191}{588}a^{4}-\frac{1793}{294}a^{3}+\frac{1103}{98}a^{2}+\frac{13}{7}a-\frac{199}{196}$, $\frac{149}{5292}a^{17}+\frac{55}{5292}a^{16}-\frac{23}{5292}a^{15}-\frac{1}{882}a^{14}+\frac{1}{1764}a^{13}-\frac{1415}{294}a^{11}-\frac{785}{441}a^{10}+\frac{661}{882}a^{9}+\frac{26}{147}a^{8}-\frac{19}{294}a^{7}-\frac{1}{21}a^{6}+\frac{161323}{588}a^{5}+\frac{19867}{196}a^{4}-\frac{25145}{588}a^{3}-\frac{485}{49}a^{2}+\frac{753}{196}a+\frac{31}{14}$
| 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: | \( 16809351.5152 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 16809351.5152 \cdot 108}{6\cdot\sqrt{450283905890997363000000000000}}\cr\approx \mathstrut & 6.88175278372 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 171*x^12 + 9747*x^6 + 27)
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 - 171*x^12 + 9747*x^6 + 27, 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 - 171*x^12 + 9747*x^6 + 27);
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 - 171*x^12 + 9747*x^6 + 27);
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 solvable group of order 18 |
| The 6 conjugacy class representatives for $C_3^2 : C_2$ |
| Character table for $C_3^2 : C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.972.2 x3, 3.1.6075.2 x3, 3.1.24300.4 x3, 3.1.2700.1 x3, 6.0.2834352.2, 6.0.110716875.2, 6.0.1771470000.2, 6.0.21870000.2, 9.1.387420489000000.19 x9 |
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
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 | R | ${\href{/padicField/7.1.0.1}{1} }^{18}$ | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| Deg $18$ | $18$ | $1$ | $37$ | |||
|
\(5\)
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |