Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 154*x^15 + 399*x^14 - 822*x^13 + 1393*x^12 - 1974*x^11 + 2358*x^10 - 2374*x^9 + 2019*x^8 - 1455*x^7 + 896*x^6 - 471*x^5 + 210*x^4 - 78*x^3 + 24*x^2 - 6*x + 1)
gp: K = bnfinit(y^18 - 9*y^17 + 45*y^16 - 154*y^15 + 399*y^14 - 822*y^13 + 1393*y^12 - 1974*y^11 + 2358*y^10 - 2374*y^9 + 2019*y^8 - 1455*y^7 + 896*y^6 - 471*y^5 + 210*y^4 - 78*y^3 + 24*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 45*x^16 - 154*x^15 + 399*x^14 - 822*x^13 + 1393*x^12 - 1974*x^11 + 2358*x^10 - 2374*x^9 + 2019*x^8 - 1455*x^7 + 896*x^6 - 471*x^5 + 210*x^4 - 78*x^3 + 24*x^2 - 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 45*x^16 - 154*x^15 + 399*x^14 - 822*x^13 + 1393*x^12 - 1974*x^11 + 2358*x^10 - 2374*x^9 + 2019*x^8 - 1455*x^7 + 896*x^6 - 471*x^5 + 210*x^4 - 78*x^3 + 24*x^2 - 6*x + 1)
\( x^{18} - 9 x^{17} + 45 x^{16} - 154 x^{15} + 399 x^{14} - 822 x^{13} + 1393 x^{12} - 1974 x^{11} + \cdots + 1 \)
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: |
\(-42845606719488000000\)
\(\medspace = -\,2^{18}\cdot 3^{21}\cdot 5^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.32\) | 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: | $2^{3/2}3^{7/6}5^{1/2}\approx 22.786176627742247$ | ||
| 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: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13}a^{16}+\frac{1}{13}a^{15}-\frac{1}{13}a^{14}+\frac{1}{13}a^{13}-\frac{3}{13}a^{12}+\frac{2}{13}a^{11}-\frac{5}{13}a^{10}-\frac{4}{13}a^{9}-\frac{2}{13}a^{8}+\frac{1}{13}a^{7}-\frac{4}{13}a^{6}-\frac{4}{13}a^{5}+\frac{1}{13}a^{4}-\frac{3}{13}a^{3}-\frac{6}{13}a^{2}+\frac{4}{13}a-\frac{3}{13}$, $\frac{1}{13949}a^{17}+\frac{8}{377}a^{16}-\frac{6245}{13949}a^{15}-\frac{6742}{13949}a^{14}-\frac{6481}{13949}a^{13}+\frac{4304}{13949}a^{12}-\frac{24}{1073}a^{11}+\frac{2655}{13949}a^{10}-\frac{2274}{13949}a^{9}-\frac{2786}{13949}a^{8}+\frac{1032}{13949}a^{7}-\frac{6449}{13949}a^{6}-\frac{3532}{13949}a^{5}+\frac{5999}{13949}a^{4}+\frac{3659}{13949}a^{3}-\frac{3222}{13949}a^{2}+\frac{1255}{13949}a-\frac{3511}{13949}$
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: |
\( -\frac{227}{377} a^{17} + \frac{1915}{377} a^{16} - \frac{9214}{377} a^{15} + \frac{30235}{377} a^{14} - \frac{75151}{377} a^{13} + \frac{147612}{377} a^{12} - \frac{237330}{377} a^{11} + \frac{24297}{29} a^{10} - \frac{350612}{377} a^{9} + \frac{24763}{29} a^{8} - \frac{245835}{377} a^{7} + \frac{157154}{377} a^{6} - \frac{88797}{377} a^{5} + \frac{44176}{377} a^{4} - \frac{19637}{377} a^{3} + \frac{6858}{377} a^{2} - \frac{2425}{377} a + \frac{802}{377} \)
(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: |
$\frac{5219}{13949}a^{17}-\frac{1371}{377}a^{16}+\frac{20448}{1073}a^{15}-\frac{73099}{1073}a^{14}+\frac{2549288}{13949}a^{13}-\frac{5405380}{13949}a^{12}+\frac{9338805}{13949}a^{11}-\frac{13387025}{13949}a^{10}+\frac{16009607}{13949}a^{9}-\frac{15952202}{13949}a^{8}+\frac{13136287}{13949}a^{7}-\frac{8946141}{13949}a^{6}+\frac{5092017}{13949}a^{5}-\frac{2522909}{13949}a^{4}+\frac{1048461}{13949}a^{3}-\frac{351506}{13949}a^{2}+\frac{83947}{13949}a-\frac{20675}{13949}$, $\frac{301}{1073}a^{17}-\frac{1292}{377}a^{16}+\frac{274505}{13949}a^{15}-\frac{1043624}{13949}a^{14}+\frac{2908032}{13949}a^{13}-\frac{6336321}{13949}a^{12}+\frac{11180878}{13949}a^{11}-\frac{16388773}{13949}a^{10}+\frac{20099663}{13949}a^{9}-\frac{20666991}{13949}a^{8}+\frac{17757594}{13949}a^{7}-\frac{12878269}{13949}a^{6}+\frac{7909680}{13949}a^{5}-\frac{4123473}{13949}a^{4}+\frac{1754983}{13949}a^{3}-\frac{628714}{13949}a^{2}+\frac{198199}{13949}a-\frac{49209}{13949}$, $\frac{4374}{13949}a^{17}-\frac{997}{377}a^{16}+\frac{171411}{13949}a^{15}-\frac{538895}{13949}a^{14}+\frac{1259395}{13949}a^{13}-\frac{2287725}{13949}a^{12}+\frac{3309300}{13949}a^{11}-\frac{3810332}{13949}a^{10}+\frac{3372673}{13949}a^{9}-\frac{2074012}{13949}a^{8}+\frac{559963}{13949}a^{7}+\frac{385378}{13949}a^{6}-\frac{567531}{13949}a^{5}+\frac{315946}{13949}a^{4}-\frac{115213}{13949}a^{3}+\frac{6192}{13949}a^{2}+\frac{9559}{13949}a-\frac{7849}{13949}$, $\frac{900}{1073}a^{17}-\frac{2680}{377}a^{16}+\frac{467263}{13949}a^{15}-\frac{1500932}{13949}a^{14}+\frac{3634388}{13949}a^{13}-\frac{6971444}{13949}a^{12}+\frac{10971371}{13949}a^{11}-\frac{14383415}{13949}a^{10}+\frac{15792722}{13949}a^{9}-\frac{14521515}{13949}a^{8}+\frac{11301840}{13949}a^{7}-\frac{7583904}{13949}a^{6}+\frac{4421882}{13949}a^{5}-\frac{2170528}{13949}a^{4}+\frac{825956}{13949}a^{3}-\frac{253992}{13949}a^{2}+\frac{71386}{13949}a-\frac{24647}{13949}$, $a$, $\frac{32144}{13949}a^{17}-\frac{6980}{377}a^{16}+\frac{1191809}{13949}a^{15}-\frac{3760830}{13949}a^{14}+\frac{9033419}{13949}a^{13}-\frac{17255465}{13949}a^{12}+\frac{27211683}{13949}a^{11}-\frac{35803672}{13949}a^{10}+\frac{39660749}{13949}a^{9}-\frac{36864592}{13949}a^{8}+\frac{29188559}{13949}a^{7}-\frac{19759862}{13949}a^{6}+\frac{11707952}{13949}a^{5}-\frac{5866284}{13949}a^{4}+\frac{2561398}{13949}a^{3}-\frac{879722}{13949}a^{2}+\frac{258805}{13949}a-\frac{40218}{13949}$, $\frac{18393}{13949}a^{17}-\frac{4439}{377}a^{16}+\frac{813699}{13949}a^{15}-\frac{2759825}{13949}a^{14}+\frac{7088140}{13949}a^{13}-\frac{14486895}{13949}a^{12}+\frac{24375129}{13949}a^{11}-\frac{34311209}{13949}a^{10}+\frac{3129765}{1073}a^{9}-\frac{40611614}{13949}a^{8}+\frac{34143066}{13949}a^{7}-\frac{24163486}{13949}a^{6}+\frac{14368179}{13949}a^{5}-\frac{7098097}{13949}a^{4}+\frac{2858826}{13949}a^{3}-\frac{921090}{13949}a^{2}+\frac{216512}{13949}a-\frac{32581}{13949}$, $\frac{14666}{13949}a^{17}-\frac{4124}{377}a^{16}+\frac{820809}{13949}a^{15}-\frac{2976651}{13949}a^{14}+\frac{8030618}{13949}a^{13}-\frac{17049950}{13949}a^{12}+\frac{29455476}{13949}a^{11}-\frac{42276067}{13949}a^{10}+\frac{50728723}{13949}a^{9}-\frac{50842668}{13949}a^{8}+\frac{42375563}{13949}a^{7}-\frac{29486416}{13949}a^{6}+\frac{17200026}{13949}a^{5}-\frac{8422351}{13949}a^{4}+\frac{3355289}{13949}a^{3}-\frac{1069786}{13949}a^{2}+\frac{277495}{13949}a-\frac{41976}{13949}$
| 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: | \( 735.644505219 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 735.644505219 \cdot 1}{6\cdot\sqrt{42845606719488000000}}\cr\approx \mathstrut & 0.285879103641 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 154*x^15 + 399*x^14 - 822*x^13 + 1393*x^12 - 1974*x^11 + 2358*x^10 - 2374*x^9 + 2019*x^8 - 1455*x^7 + 896*x^6 - 471*x^5 + 210*x^4 - 78*x^3 + 24*x^2 - 6*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 - 9*x^17 + 45*x^16 - 154*x^15 + 399*x^14 - 822*x^13 + 1393*x^12 - 1974*x^11 + 2358*x^10 - 2374*x^9 + 2019*x^8 - 1455*x^7 + 896*x^6 - 471*x^5 + 210*x^4 - 78*x^3 + 24*x^2 - 6*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 - 9*x^17 + 45*x^16 - 154*x^15 + 399*x^14 - 822*x^13 + 1393*x^12 - 1974*x^11 + 2358*x^10 - 2374*x^9 + 2019*x^8 - 1455*x^7 + 896*x^6 - 471*x^5 + 210*x^4 - 78*x^3 + 24*x^2 - 6*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 - 9*x^17 + 45*x^16 - 154*x^15 + 399*x^14 - 822*x^13 + 1393*x^12 - 1974*x^11 + 2358*x^10 - 2374*x^9 + 2019*x^8 - 1455*x^7 + 896*x^6 - 471*x^5 + 210*x^4 - 78*x^3 + 24*x^2 - 6*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
$S_3\times D_6$ (as 18T29):
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 72 |
| The 18 conjugacy class representatives for $S_3\times D_6$ |
| Character table for $S_3\times D_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.216.1, 3.1.135.1, 6.0.139968.1, 6.0.54675.1, 9.1.3779136000.1 |
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 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.4.26873856000000.1 |
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 | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
|
\(3\)
| 3.6.7.5 | $x^{6} + 6 x^{2} + 3$ | $6$ | $1$ | $7$ | $D_{6}$ | $[3/2]_{2}^{2}$ |
| 3.12.14.11 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 792 x^{8} + 1728 x^{7} + 2918 x^{6} + 3684 x^{5} + 3156 x^{4} + 1376 x^{3} - 36 x^{2} - 168 x + 25$ | $6$ | $2$ | $14$ | $D_6$ | $[3/2]_{2}^{2}$ | |
|
\(5\)
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |