Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^15 + 46*x^12 - 171*x^9 + 388*x^6 - 171*x^3 + 27)
gp: K = bnfinit(y^18 - 9*y^15 + 46*y^12 - 171*y^9 + 388*y^6 - 171*y^3 + 27, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^15 + 46*x^12 - 171*x^9 + 388*x^6 - 171*x^3 + 27);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^15 + 46*x^12 - 171*x^9 + 388*x^6 - 171*x^3 + 27)
\( x^{18} - 9x^{15} + 46x^{12} - 171x^{9} + 388x^{6} - 171x^{3} + 27 \)
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: |
\(-7625597484987000000000000\)
\(\medspace = -\,2^{12}\cdot 3^{27}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.12\) | 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: | $2^{2/3}3^{3/2}5^{2/3}\approx 24.11840306298509$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11}a^{12}-\frac{5}{11}a^{9}-\frac{5}{11}a^{6}-\frac{3}{11}a^{3}+\frac{3}{11}$, $\frac{1}{11}a^{13}-\frac{5}{11}a^{10}-\frac{5}{11}a^{7}-\frac{3}{11}a^{4}+\frac{3}{11}a$, $\frac{1}{33}a^{14}+\frac{2}{11}a^{11}-\frac{5}{33}a^{8}-\frac{1}{11}a^{5}-\frac{8}{33}a^{2}$, $\frac{1}{89133}a^{15}-\frac{741}{29711}a^{12}+\frac{35659}{89133}a^{9}+\frac{10248}{29711}a^{6}+\frac{1051}{2409}a^{3}-\frac{14043}{29711}$, $\frac{1}{267399}a^{16}-\frac{247}{29711}a^{13}+\frac{35659}{267399}a^{10}+\frac{3416}{29711}a^{7}-\frac{1358}{7227}a^{4}-\frac{4681}{29711}a$, $\frac{1}{267399}a^{17}-\frac{247}{29711}a^{14}+\frac{35659}{267399}a^{11}+\frac{3416}{29711}a^{8}-\frac{1358}{7227}a^{5}-\frac{4681}{29711}a^{2}$
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: |
\( \frac{12293}{267399} a^{17} - \frac{3314}{8103} a^{14} + \frac{550997}{267399} a^{11} - \frac{673927}{89133} a^{8} + \frac{120707}{7227} a^{5} - \frac{452653}{89133} a^{2} \)
(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: |
$\frac{119}{29711}a^{17}+\frac{172}{8103}a^{15}-\frac{2218}{89133}a^{14}-\frac{505}{2701}a^{12}+\frac{2851}{29711}a^{11}+\frac{7480}{8103}a^{9}-\frac{22864}{89133}a^{8}-\frac{9200}{2701}a^{6}+\frac{93}{803}a^{5}+\frac{1630}{219}a^{3}+\frac{198989}{89133}a^{2}-\frac{3403}{2701}$, $\frac{119}{29711}a^{17}-\frac{5774}{267399}a^{16}+\frac{172}{8103}a^{15}-\frac{2218}{89133}a^{14}+\frac{5452}{29711}a^{13}-\frac{505}{2701}a^{12}+\frac{2851}{29711}a^{11}-\frac{240926}{267399}a^{10}+\frac{7480}{8103}a^{9}-\frac{22864}{89133}a^{8}+\frac{95954}{29711}a^{7}-\frac{9200}{2701}a^{6}+\frac{93}{803}a^{5}-\frac{47507}{7227}a^{4}+\frac{1630}{219}a^{3}+\frac{198989}{89133}a^{2}+\frac{7290}{29711}a-\frac{6104}{2701}$, $\frac{1373}{267399}a^{16}-\frac{1506}{29711}a^{13}+\frac{74408}{267399}a^{10}-\frac{2589}{2701}a^{7}+\frac{13829}{7227}a^{4}-\frac{6736}{29711}a$, $\frac{356}{89133}a^{15}-\frac{1799}{29711}a^{12}+\frac{29615}{89133}a^{9}-\frac{3507}{2701}a^{6}+\frac{9302}{2409}a^{3}-\frac{53777}{29711}$, $\frac{11447}{267399}a^{17}-\frac{1742}{29711}a^{16}+\frac{298}{89133}a^{15}-\frac{36200}{89133}a^{14}+\frac{15438}{29711}a^{13}+\frac{664}{29711}a^{12}+\frac{550952}{267399}a^{11}-\frac{78709}{29711}a^{10}-\frac{4754}{89133}a^{9}-\frac{684329}{89133}a^{8}+\frac{282985}{29711}a^{7}+\frac{1389}{2701}a^{6}+\frac{128366}{7227}a^{5}-\frac{1573}{73}a^{4}-\frac{3257}{2409}a^{3}-\frac{583622}{89133}a^{2}+\frac{107887}{29711}a+\frac{44952}{29711}$, $\frac{14425}{267399}a^{17}+\frac{420}{29711}a^{16}-\frac{2273}{89133}a^{15}-\frac{13861}{29711}a^{14}-\frac{4516}{29711}a^{13}+\frac{6972}{29711}a^{12}+\frac{634669}{267399}a^{11}+\frac{21343}{29711}a^{10}-\frac{95834}{89133}a^{9}-\frac{260540}{29711}a^{8}-\frac{82031}{29711}a^{7}+\frac{126667}{29711}a^{6}+\frac{137932}{7227}a^{5}+\frac{4734}{803}a^{4}-\frac{20000}{2409}a^{3}-\frac{22447}{2701}a^{2}+\frac{8174}{29711}a+\frac{88454}{29711}$, $\frac{12704}{267399}a^{17}-\frac{6206}{89133}a^{16}-\frac{19}{1221}a^{15}-\frac{12831}{29711}a^{14}+\frac{17750}{29711}a^{13}+\frac{56}{407}a^{12}+\frac{597137}{267399}a^{11}-\frac{258017}{89133}a^{10}-\frac{754}{1221}a^{9}-\frac{22354}{2701}a^{8}+\frac{306572}{29711}a^{7}+\frac{69}{37}a^{6}+\frac{132188}{7227}a^{5}-\frac{53027}{2409}a^{4}-\frac{124}{33}a^{3}-\frac{177773}{29711}a^{2}+\frac{10103}{2701}a+\frac{898}{407}$, $\frac{38902}{267399}a^{17}-\frac{18478}{267399}a^{16}-\frac{322}{29711}a^{15}-\frac{10432}{8103}a^{14}+\frac{18283}{29711}a^{13}+\frac{2742}{29711}a^{12}+\frac{1739272}{267399}a^{11}-\frac{838063}{267399}a^{10}-\frac{13752}{29711}a^{9}-\frac{2128292}{89133}a^{8}+\frac{341848}{29711}a^{7}+\frac{53617}{29711}a^{6}+\frac{380857}{7227}a^{5}-\frac{186922}{7227}a^{4}-\frac{2768}{803}a^{3}-\frac{1517189}{89133}a^{2}+\frac{303907}{29711}a+\frac{17322}{29711}$
| 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: | \( 2586019.54261 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 2586019.54261 \cdot 1}{18\cdot\sqrt{7625597484987000000000000}}\cr\approx \mathstrut & 0.794037793636 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^15 + 46*x^12 - 171*x^9 + 388*x^6 - 171*x^3 + 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 - 9*x^15 + 46*x^12 - 171*x^9 + 388*x^6 - 171*x^3 + 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 - 9*x^15 + 46*x^12 - 171*x^9 + 388*x^6 - 171*x^3 + 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 - 9*x^15 + 46*x^12 - 171*x^9 + 388*x^6 - 171*x^3 + 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
$C_3\times S_3$ (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
| \(\Q(\sqrt{-3}) \), 3.1.300.1 x3, \(\Q(\zeta_{9})^+\), 6.0.270000.1, 6.0.196830000.1 x2, \(\Q(\zeta_{9})\), 9.3.1594323000000.6 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: | 6.0.196830000.1 |
| Degree 9 sibling: | 9.3.1594323000000.6 |
| Minimal sibling: | 6.0.196830000.1 |
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 | R | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.1.0.1}{1} }^{18}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
|
\(5\)
| 5.18.12.1 | $x^{18} + 3 x^{16} + 42 x^{15} + 6 x^{14} + 54 x^{13} + 196 x^{12} - 114 x^{11} + 216 x^{10} - 3044 x^{9} - 516 x^{8} - 2670 x^{7} + 9991 x^{6} + 5898 x^{5} - 7887 x^{4} - 38142 x^{3} + 23322 x^{2} - 14520 x + 100948$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |