Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + x^16 + 2*x^15 - 9*x^14 + 6*x^13 + 3*x^12 - 21*x^11 + 32*x^10 - 24*x^9 + 14*x^8 - 5*x^7 + 13*x^6 - 14*x^5 + 6*x^4 - 3*x^3 - x^2 + 1)
gp: K = bnfinit(y^18 - y^17 + y^16 + 2*y^15 - 9*y^14 + 6*y^13 + 3*y^12 - 21*y^11 + 32*y^10 - 24*y^9 + 14*y^8 - 5*y^7 + 13*y^6 - 14*y^5 + 6*y^4 - 3*y^3 - y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 + x^16 + 2*x^15 - 9*x^14 + 6*x^13 + 3*x^12 - 21*x^11 + 32*x^10 - 24*x^9 + 14*x^8 - 5*x^7 + 13*x^6 - 14*x^5 + 6*x^4 - 3*x^3 - x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 + x^16 + 2*x^15 - 9*x^14 + 6*x^13 + 3*x^12 - 21*x^11 + 32*x^10 - 24*x^9 + 14*x^8 - 5*x^7 + 13*x^6 - 14*x^5 + 6*x^4 - 3*x^3 - x^2 + 1)
\( x^{18} - x^{17} + x^{16} + 2 x^{15} - 9 x^{14} + 6 x^{13} + 3 x^{12} - 21 x^{11} + 32 x^{10} - 24 x^{9} + \cdots + 1 \)
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: |
\(-137081836567277622747\)
\(\medspace = -\,3^{9}\cdot 19^{6}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.14\) | 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: | $3^{1/2}19^{2/3}23^{1/2}\approx 59.14621341619825$ | ||
| Ramified primes: |
\(3\), \(19\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{33}a^{14}-\frac{5}{33}a^{13}+\frac{1}{11}a^{12}+\frac{1}{3}a^{11}-\frac{1}{11}a^{10}+\frac{5}{11}a^{9}+\frac{5}{11}a^{7}+\frac{4}{11}a^{6}+\frac{14}{33}a^{5}-\frac{2}{11}a^{4}-\frac{4}{11}a^{3}-\frac{5}{11}a^{2}-\frac{1}{11}a-\frac{1}{33}$, $\frac{1}{33}a^{15}+\frac{4}{33}a^{12}-\frac{14}{33}a^{11}-\frac{1}{3}a^{10}-\frac{2}{33}a^{9}-\frac{7}{33}a^{8}-\frac{4}{11}a^{7}-\frac{14}{33}a^{6}+\frac{3}{11}a^{5}-\frac{3}{11}a^{4}+\frac{2}{33}a^{3}-\frac{1}{33}a^{2}+\frac{2}{11}a-\frac{5}{33}$, $\frac{1}{33}a^{16}+\frac{4}{33}a^{13}-\frac{1}{11}a^{12}+\frac{1}{3}a^{11}-\frac{2}{33}a^{10}+\frac{4}{33}a^{9}-\frac{1}{33}a^{8}+\frac{8}{33}a^{7}+\frac{3}{11}a^{6}+\frac{13}{33}a^{5}-\frac{3}{11}a^{4}-\frac{1}{33}a^{3}-\frac{5}{33}a^{2}-\frac{16}{33}a+\frac{1}{3}$, $\frac{1}{363}a^{17}-\frac{5}{363}a^{16}-\frac{1}{363}a^{15}-\frac{5}{363}a^{14}-\frac{1}{11}a^{13}+\frac{28}{363}a^{12}+\frac{48}{121}a^{11}+\frac{54}{121}a^{10}+\frac{5}{11}a^{9}-\frac{79}{363}a^{8}+\frac{2}{11}a^{7}-\frac{5}{363}a^{6}+\frac{2}{11}a^{5}-\frac{23}{121}a^{4}-\frac{4}{363}a^{3}+\frac{19}{121}a^{2}-\frac{53}{363}a-\frac{32}{121}$
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: |
\( \frac{586}{363} a^{17} - \frac{158}{363} a^{16} + \frac{197}{121} a^{15} + \frac{1459}{363} a^{14} - \frac{123}{11} a^{13} + \frac{799}{363} a^{12} + \frac{1235}{363} a^{11} - \frac{10415}{363} a^{10} + \frac{346}{11} a^{9} - \frac{2807}{121} a^{8} + \frac{575}{33} a^{7} - \frac{2050}{363} a^{6} + \frac{746}{33} a^{5} - \frac{3089}{363} a^{4} + \frac{2650}{363} a^{3} - \frac{416}{121} a^{2} - \frac{808}{363} a - \frac{195}{121} \)
(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: |
$\frac{419}{363}a^{17}-\frac{49}{363}a^{16}+\frac{95}{121}a^{15}+\frac{1117}{363}a^{14}-\frac{257}{33}a^{13}-\frac{75}{121}a^{12}+\frac{1717}{363}a^{11}-\frac{7318}{363}a^{10}+\frac{578}{33}a^{9}-\frac{2807}{363}a^{8}+\frac{167}{33}a^{7}-\frac{5}{363}a^{6}+\frac{499}{33}a^{5}-\frac{938}{363}a^{4}+\frac{524}{363}a^{3}-\frac{262}{363}a^{2}-\frac{113}{121}a-\frac{613}{363}$, $\frac{1}{363}a^{17}-\frac{16}{363}a^{16}+\frac{10}{363}a^{15}-\frac{16}{363}a^{14}-\frac{2}{33}a^{13}+\frac{193}{363}a^{12}-\frac{10}{363}a^{11}+\frac{32}{121}a^{10}+\frac{38}{33}a^{9}-\frac{250}{121}a^{8}+\frac{26}{33}a^{7}-\frac{130}{121}a^{6}-\frac{56}{33}a^{5}+\frac{239}{363}a^{4}-\frac{202}{363}a^{3}+\frac{145}{363}a^{2}+\frac{101}{363}a+\frac{586}{363}$, $\frac{70}{121}a^{17}-\frac{71}{363}a^{16}+\frac{241}{363}a^{15}+\frac{523}{363}a^{14}-\frac{131}{33}a^{13}+\frac{512}{363}a^{12}+\frac{331}{363}a^{11}-\frac{3754}{363}a^{10}+\frac{133}{11}a^{9}-\frac{3907}{363}a^{8}+\frac{248}{33}a^{7}-\frac{1094}{363}a^{6}+\frac{256}{33}a^{5}-\frac{982}{363}a^{4}+\frac{898}{363}a^{3}-\frac{394}{363}a^{2}-\frac{251}{363}a-\frac{10}{121}$, $\frac{229}{363}a^{17}-\frac{265}{363}a^{16}+\frac{244}{363}a^{15}+\frac{439}{363}a^{14}-\frac{196}{33}a^{13}+\frac{1583}{363}a^{12}+\frac{581}{363}a^{11}-\frac{1703}{121}a^{10}+\frac{703}{33}a^{9}-\frac{6068}{363}a^{8}+\frac{112}{11}a^{7}-\frac{510}{121}a^{6}+\frac{340}{33}a^{5}-\frac{3602}{363}a^{4}+\frac{1559}{363}a^{3}-\frac{71}{121}a^{2}-\frac{59}{363}a-\frac{13}{121}$, $\frac{8}{33}a^{17}+\frac{1}{3}a^{16}-\frac{8}{33}a^{15}+\frac{9}{11}a^{14}-\frac{32}{33}a^{13}-\frac{113}{33}a^{12}+\frac{32}{11}a^{11}-\frac{21}{11}a^{10}-\frac{100}{33}a^{9}+\frac{95}{11}a^{8}-\frac{182}{33}a^{7}+\frac{167}{33}a^{6}-\frac{20}{11}a^{5}+\frac{193}{33}a^{4}-\frac{161}{33}a^{3}-\frac{56}{33}a^{2}-a+\frac{34}{33}$, $\frac{15}{121}a^{17}+\frac{57}{121}a^{16}-\frac{23}{363}a^{15}+\frac{314}{363}a^{14}+\frac{2}{11}a^{13}-\frac{416}{121}a^{12}+\frac{485}{363}a^{11}-\frac{685}{363}a^{10}-\frac{188}{33}a^{9}+\frac{3397}{363}a^{8}-\frac{94}{11}a^{7}+\frac{1069}{121}a^{6}-\frac{29}{11}a^{5}+\frac{868}{121}a^{4}-\frac{950}{363}a^{3}+\frac{706}{363}a^{2}-\frac{1010}{363}a-\frac{250}{363}$, $\frac{217}{363}a^{17}+\frac{92}{363}a^{16}+\frac{34}{121}a^{15}+\frac{763}{363}a^{14}-\frac{106}{33}a^{13}-\frac{920}{363}a^{12}+\frac{1372}{363}a^{11}-\frac{3841}{363}a^{10}+\frac{95}{33}a^{9}+\frac{1469}{363}a^{8}-\frac{60}{11}a^{7}+\frac{1808}{363}a^{6}+\frac{202}{33}a^{5}+\frac{889}{363}a^{4}-\frac{238}{121}a^{3}+\frac{1072}{363}a^{2}-\frac{809}{363}a-\frac{559}{363}$, $\frac{119}{121}a^{17}-\frac{122}{121}a^{16}+\frac{380}{363}a^{15}+\frac{778}{363}a^{14}-\frac{290}{33}a^{13}+\frac{780}{121}a^{12}+\frac{1160}{363}a^{11}-\frac{2623}{121}a^{10}+\frac{1048}{33}a^{9}-\frac{8887}{363}a^{8}+\frac{398}{33}a^{7}-\frac{1367}{363}a^{6}+\frac{391}{33}a^{5}-\frac{4844}{363}a^{4}+\frac{745}{121}a^{3}-\frac{155}{363}a^{2}-\frac{364}{363}a+\frac{5}{121}$
| 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: | \( 955.275046862 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 955.275046862 \cdot 1}{6\cdot\sqrt{137081836567277622747}}\cr\approx \mathstrut & 0.207542020862 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + x^16 + 2*x^15 - 9*x^14 + 6*x^13 + 3*x^12 - 21*x^11 + 32*x^10 - 24*x^9 + 14*x^8 - 5*x^7 + 13*x^6 - 14*x^5 + 6*x^4 - 3*x^3 - x^2 + 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 - x^17 + x^16 + 2*x^15 - 9*x^14 + 6*x^13 + 3*x^12 - 21*x^11 + 32*x^10 - 24*x^9 + 14*x^8 - 5*x^7 + 13*x^6 - 14*x^5 + 6*x^4 - 3*x^3 - x^2 + 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 - x^17 + x^16 + 2*x^15 - 9*x^14 + 6*x^13 + 3*x^12 - 21*x^11 + 32*x^10 - 24*x^9 + 14*x^8 - 5*x^7 + 13*x^6 - 14*x^5 + 6*x^4 - 3*x^3 - x^2 + 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 - x^17 + x^16 + 2*x^15 - 9*x^14 + 6*x^13 + 3*x^12 - 21*x^11 + 32*x^10 - 24*x^9 + 14*x^8 - 5*x^7 + 13*x^6 - 14*x^5 + 6*x^4 - 3*x^3 - x^2 + 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
$C_3\times S_3^2$ (as 18T46):
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 108 |
| The 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.23.1, 6.0.9747.1, 6.0.14283.1 |
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]
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.14064002930879001.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | R | R | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(23\)
| 23.6.0.1 | $x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 23.12.6.1 | $x^{12} + 140 x^{10} + 18 x^{9} + 8092 x^{8} + 20 x^{7} + 244001 x^{6} - 56140 x^{5} + 4059563 x^{4} - 1721304 x^{3} + 36312468 x^{2} - 14694000 x + 141161628$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |