Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - x^15 - 5*x^14 - 5*x^12 + 10*x^11 - 11*x^10 + 19*x^9 - 8*x^8 - 17*x^7 + 46*x^6 - 57*x^5 + 52*x^4 - 37*x^3 + 14*x^2 - 3*x + 1)
gp: K = bnfinit(y^18 + 2*y^16 - y^15 - 5*y^14 - 5*y^12 + 10*y^11 - 11*y^10 + 19*y^9 - 8*y^8 - 17*y^7 + 46*y^6 - 57*y^5 + 52*y^4 - 37*y^3 + 14*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 2*x^16 - x^15 - 5*x^14 - 5*x^12 + 10*x^11 - 11*x^10 + 19*x^9 - 8*x^8 - 17*x^7 + 46*x^6 - 57*x^5 + 52*x^4 - 37*x^3 + 14*x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 2*x^16 - x^15 - 5*x^14 - 5*x^12 + 10*x^11 - 11*x^10 + 19*x^9 - 8*x^8 - 17*x^7 + 46*x^6 - 57*x^5 + 52*x^4 - 37*x^3 + 14*x^2 - 3*x + 1)
\( x^{18} + 2 x^{16} - x^{15} - 5 x^{14} - 5 x^{12} + 10 x^{11} - 11 x^{10} + 19 x^{9} - 8 x^{8} - 17 x^{7} + \cdots + 1 \)
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: |
\(-16481672572799864375\)
\(\medspace = -\,5^{4}\cdot 11^{4}\cdot 23^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.68\) | 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: | $5^{2/3}11^{2/3}23^{1/2}\approx 69.35946123538706$ | ||
| Ramified primes: |
\(5\), \(11\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5}a^{15}+\frac{1}{5}a^{14}+\frac{1}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{2}{5}a^{10}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{16}+\frac{1}{5}a^{12}+\frac{1}{5}a^{11}-\frac{1}{5}a^{10}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}$, $\frac{1}{3029771545}a^{17}-\frac{292269546}{3029771545}a^{16}+\frac{113494807}{3029771545}a^{15}-\frac{676258248}{3029771545}a^{14}-\frac{624457397}{3029771545}a^{13}-\frac{1001474408}{3029771545}a^{12}-\frac{1233781428}{3029771545}a^{11}+\frac{701788097}{3029771545}a^{10}-\frac{217782827}{3029771545}a^{9}-\frac{804115889}{3029771545}a^{8}+\frac{257956867}{3029771545}a^{7}-\frac{138295421}{605954309}a^{6}+\frac{160462277}{605954309}a^{5}+\frac{490679776}{3029771545}a^{4}+\frac{23624492}{605954309}a^{3}-\frac{1190301754}{3029771545}a^{2}+\frac{1455677013}{3029771545}a+\frac{1387349104}{3029771545}$
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: |
\( -1 \)
(order $2$)
| 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: |
$\frac{1970603636}{3029771545}a^{17}+\frac{2188112856}{3029771545}a^{16}+\frac{5530995419}{3029771545}a^{15}+\frac{3399328134}{3029771545}a^{14}-\frac{1624862711}{605954309}a^{13}-\frac{9996481564}{3029771545}a^{12}-\frac{16986702342}{3029771545}a^{11}+\frac{4244500771}{3029771545}a^{10}-\frac{11383889953}{3029771545}a^{9}+\frac{21155039043}{3029771545}a^{8}+\frac{12489846706}{3029771545}a^{7}-\frac{28224723489}{3029771545}a^{6}+\frac{11064414150}{605954309}a^{5}-\frac{35883170422}{3029771545}a^{4}+\frac{36276346071}{3029771545}a^{3}-\frac{16368322243}{3029771545}a^{2}-\frac{4338763561}{3029771545}a-\frac{4133725821}{3029771545}$, $\frac{1479144098}{3029771545}a^{17}+\frac{1522240673}{3029771545}a^{16}+\frac{3956095197}{3029771545}a^{15}+\frac{2206770102}{3029771545}a^{14}-\frac{1315669487}{605954309}a^{13}-\frac{7394147687}{3029771545}a^{12}-\frac{12609467876}{3029771545}a^{11}+\frac{3322782878}{3029771545}a^{10}-\frac{8287992694}{3029771545}a^{9}+\frac{18219348324}{3029771545}a^{8}+\frac{11971924238}{3029771545}a^{7}-\frac{21299829087}{3029771545}a^{6}+\frac{8603570360}{605954309}a^{5}-\frac{30284280881}{3029771545}a^{4}+\frac{23658661313}{3029771545}a^{3}-\frac{9454765954}{3029771545}a^{2}-\frac{3560487678}{3029771545}a-\frac{2788157243}{3029771545}$, $\frac{24336452}{605954309}a^{17}+\frac{3298373}{605954309}a^{16}-\frac{8479036}{605954309}a^{15}-\frac{88164388}{605954309}a^{14}-\frac{323042460}{605954309}a^{13}-\frac{185736271}{605954309}a^{12}+\frac{31614020}{605954309}a^{11}+\frac{477455930}{605954309}a^{10}+\frac{457744720}{605954309}a^{9}+\frac{607510476}{605954309}a^{8}+\frac{403693276}{605954309}a^{7}-\frac{1072338463}{605954309}a^{6}+\frac{706054491}{605954309}a^{5}-\frac{411748723}{605954309}a^{4}-\frac{809334002}{605954309}a^{3}+\frac{590251126}{605954309}a^{2}-\frac{872556126}{605954309}a+\frac{64562806}{605954309}$, $\frac{2231209431}{3029771545}a^{17}+\frac{213070895}{605954309}a^{16}+\frac{5099750773}{3029771545}a^{15}+\frac{339307578}{3029771545}a^{14}-\frac{10812281816}{3029771545}a^{13}-\frac{4953283861}{3029771545}a^{12}-\frac{2864145163}{605954309}a^{11}+\frac{14952559549}{3029771545}a^{10}-\frac{3576565377}{605954309}a^{9}+\frac{33840404492}{3029771545}a^{8}-\frac{2850765336}{3029771545}a^{7}-\frac{39048644127}{3029771545}a^{6}+\frac{17490063547}{605954309}a^{5}-\frac{89167593903}{3029771545}a^{4}+\frac{78158916648}{3029771545}a^{3}-\frac{43711182016}{3029771545}a^{2}+\frac{6987847816}{3029771545}a-\frac{661590491}{3029771545}$, $\frac{2141844549}{3029771545}a^{17}+\frac{474171084}{605954309}a^{16}+\frac{6169538819}{3029771545}a^{15}+\frac{3850939419}{3029771545}a^{14}-\frac{8413245237}{3029771545}a^{13}-\frac{10345056262}{3029771545}a^{12}-\frac{18831684471}{3029771545}a^{11}+\frac{4735385992}{3029771545}a^{10}-\frac{12635953821}{3029771545}a^{9}+\frac{23499525277}{3029771545}a^{8}+\frac{11949484773}{3029771545}a^{7}-\frac{31054415754}{3029771545}a^{6}+\frac{59312621386}{3029771545}a^{5}-\frac{44614309672}{3029771545}a^{4}+\frac{44298016219}{3029771545}a^{3}-\frac{17286913153}{3029771545}a^{2}-\frac{446451261}{605954309}a-\frac{2611432288}{3029771545}$, $\frac{62798952}{3029771545}a^{17}+\frac{1519346228}{3029771545}a^{16}+\frac{317507940}{605954309}a^{15}+\frac{788819537}{605954309}a^{14}+\frac{1629911697}{3029771545}a^{13}-\frac{1335467913}{605954309}a^{12}-\frac{6864065139}{3029771545}a^{11}-\frac{10695378143}{3029771545}a^{10}+\frac{5731601328}{3029771545}a^{9}-\frac{6789354101}{3029771545}a^{8}+\frac{3312512846}{605954309}a^{7}+\frac{4382689582}{3029771545}a^{6}-\frac{21923343522}{3029771545}a^{5}+\frac{41750092592}{3029771545}a^{4}-\frac{32638152494}{3029771545}a^{3}+\frac{5651008404}{605954309}a^{2}-\frac{11211426441}{3029771545}a-\frac{3189513834}{3029771545}$, $\frac{2053140866}{3029771545}a^{17}+\frac{2051356828}{3029771545}a^{16}+\frac{6086776662}{3029771545}a^{15}+\frac{3405588862}{3029771545}a^{14}-\frac{7837855387}{3029771545}a^{13}-\frac{9410546054}{3029771545}a^{12}-\frac{20281477869}{3029771545}a^{11}+\frac{3564272443}{3029771545}a^{10}-\frac{14071925587}{3029771545}a^{9}+\frac{29569204437}{3029771545}a^{8}+\frac{10828300606}{3029771545}a^{7}-\frac{25415081141}{3029771545}a^{6}+\frac{62622551198}{3029771545}a^{5}-\frac{11777547045}{605954309}a^{4}+\frac{55067368568}{3029771545}a^{3}-\frac{29185639114}{3029771545}a^{2}+\frac{5934806303}{3029771545}a-\frac{6168377713}{3029771545}$, $\frac{2023547642}{3029771545}a^{17}+\frac{1687271703}{3029771545}a^{16}+\frac{1143763339}{605954309}a^{15}+\frac{505362088}{605954309}a^{14}-\frac{7573673053}{3029771545}a^{13}-\frac{1345506649}{605954309}a^{12}-\frac{16738283774}{3029771545}a^{11}+\frac{8036592142}{3029771545}a^{10}-\frac{16004106962}{3029771545}a^{9}+\frac{27395991984}{3029771545}a^{8}+\frac{241426321}{605954309}a^{7}-\frac{28001707463}{3029771545}a^{6}+\frac{65557409613}{3029771545}a^{5}-\frac{66903634313}{3029771545}a^{4}+\frac{67138368436}{3029771545}a^{3}-\frac{7987060061}{605954309}a^{2}+\frac{10779227694}{3029771545}a-\frac{4710336964}{3029771545}$
| 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: | \( 102.074554739 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 102.074554739 \cdot 1}{2\cdot\sqrt{16481672572799864375}}\cr\approx \mathstrut & 0.191869380201 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - x^15 - 5*x^14 - 5*x^12 + 10*x^11 - 11*x^10 + 19*x^9 - 8*x^8 - 17*x^7 + 46*x^6 - 57*x^5 + 52*x^4 - 37*x^3 + 14*x^2 - 3*x + 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 + 2*x^16 - x^15 - 5*x^14 - 5*x^12 + 10*x^11 - 11*x^10 + 19*x^9 - 8*x^8 - 17*x^7 + 46*x^6 - 57*x^5 + 52*x^4 - 37*x^3 + 14*x^2 - 3*x + 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 + 2*x^16 - x^15 - 5*x^14 - 5*x^12 + 10*x^11 - 11*x^10 + 19*x^9 - 8*x^8 - 17*x^7 + 46*x^6 - 57*x^5 + 52*x^4 - 37*x^3 + 14*x^2 - 3*x + 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 + 2*x^16 - x^15 - 5*x^14 - 5*x^12 + 10*x^11 - 11*x^10 + 19*x^9 - 8*x^8 - 17*x^7 + 46*x^6 - 57*x^5 + 52*x^4 - 37*x^3 + 14*x^2 - 3*x + 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
$C_3^3:S_3$ (as 18T88):
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 162 |
| The 13 conjugacy class representatives for $C_3^3:S_3$ |
| Character table for $C_3^3:S_3$ |
Intermediate fields
| \(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.846519025.1 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 siblings: | data not computed |
| Minimal sibling: | 9.1.846519025.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}$ | R | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(11\)
| 11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 11.6.4.1 | $x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |