Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 54*x^15 + 116*x^14 - 226*x^13 + 412*x^12 - 686*x^11 + 997*x^10 - 1220*x^9 + 1225*x^8 - 994*x^7 + 652*x^6 - 356*x^5 + 169*x^4 - 70*x^3 + 25*x^2 - 6*x + 1)
gp: K = bnfinit(y^18 - 6*y^17 + 21*y^16 - 54*y^15 + 116*y^14 - 226*y^13 + 412*y^12 - 686*y^11 + 997*y^10 - 1220*y^9 + 1225*y^8 - 994*y^7 + 652*y^6 - 356*y^5 + 169*y^4 - 70*y^3 + 25*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 21*x^16 - 54*x^15 + 116*x^14 - 226*x^13 + 412*x^12 - 686*x^11 + 997*x^10 - 1220*x^9 + 1225*x^8 - 994*x^7 + 652*x^6 - 356*x^5 + 169*x^4 - 70*x^3 + 25*x^2 - 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 21*x^16 - 54*x^15 + 116*x^14 - 226*x^13 + 412*x^12 - 686*x^11 + 997*x^10 - 1220*x^9 + 1225*x^8 - 994*x^7 + 652*x^6 - 356*x^5 + 169*x^4 - 70*x^3 + 25*x^2 - 6*x + 1)
\( x^{18} - 6 x^{17} + 21 x^{16} - 54 x^{15} + 116 x^{14} - 226 x^{13} + 412 x^{12} - 686 x^{11} + 997 x^{10} + \cdots + 1 \)
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: |
\(-56693912375296000000\)
\(\medspace = -\,2^{18}\cdot 5^{6}\cdot 7^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.51\) | 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: | $2\cdot 5^{1/2}7^{2/3}\approx 16.364912636128995$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{256977029}a^{17}-\frac{40852282}{256977029}a^{16}-\frac{90896428}{256977029}a^{15}-\frac{71267825}{256977029}a^{14}-\frac{1410251}{256977029}a^{13}+\frac{25972511}{256977029}a^{12}+\frac{121908911}{256977029}a^{11}-\frac{26733424}{256977029}a^{10}-\frac{63355470}{256977029}a^{9}+\frac{73314996}{256977029}a^{8}+\frac{38926272}{256977029}a^{7}-\frac{14676092}{256977029}a^{6}-\frac{60488711}{256977029}a^{5}+\frac{126710720}{256977029}a^{4}+\frac{103649471}{256977029}a^{3}+\frac{98776940}{256977029}a^{2}-\frac{69678780}{256977029}a-\frac{111122987}{256977029}$
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: |
\( -\frac{1958110}{2274133} a^{17} + \frac{12793711}{2274133} a^{16} - \frac{46575486}{2274133} a^{15} + \frac{122807849}{2274133} a^{14} - \frac{266884792}{2274133} a^{13} + \frac{521638226}{2274133} a^{12} - \frac{954283917}{2274133} a^{11} + \frac{1603216417}{2274133} a^{10} - \frac{2360935244}{2274133} a^{9} + \frac{2923198932}{2274133} a^{8} - \frac{2951830859}{2274133} a^{7} + \frac{2374393719}{2274133} a^{6} - \frac{1506363920}{2274133} a^{5} + \frac{772674530}{2274133} a^{4} - \frac{344556129}{2274133} a^{3} + \frac{137069085}{2274133} a^{2} - \frac{42484885}{2274133} a + \frac{8138860}{2274133} \)
(order $4$)
| 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: |
$\frac{322605173}{256977029}a^{17}-\frac{1912714781}{256977029}a^{16}+\frac{6574766725}{256977029}a^{15}-\frac{16610647877}{256977029}a^{14}+\frac{35125982382}{256977029}a^{13}-\frac{67755434956}{256977029}a^{12}+\frac{122651129615}{256977029}a^{11}-\frac{202317220541}{256977029}a^{10}+\frac{288882835453}{256977029}a^{9}-\frac{343597336850}{256977029}a^{8}+\frac{330700848807}{256977029}a^{7}-\frac{252735170102}{256977029}a^{6}+\frac{153450118754}{256977029}a^{5}-\frac{77478873499}{256977029}a^{4}+\frac{34833122905}{256977029}a^{3}-\frac{13368163194}{256977029}a^{2}+\frac{4117642201}{256977029}a-\frac{600460198}{256977029}$, $\frac{57797732}{256977029}a^{17}-\frac{316892203}{256977029}a^{16}+\frac{929262311}{256977029}a^{15}-\frac{1986580304}{256977029}a^{14}+\frac{3547271039}{256977029}a^{13}-\frac{6067346493}{256977029}a^{12}+\frac{10033825374}{256977029}a^{11}-\frac{14230745054}{256977029}a^{10}+\frac{14055942983}{256977029}a^{9}-\frac{4843284980}{256977029}a^{8}-\frac{12176530115}{256977029}a^{7}+\frac{27346309641}{256977029}a^{6}-\frac{30339588892}{256977029}a^{5}+\frac{21618031041}{256977029}a^{4}-\frac{11224252797}{256977029}a^{3}+\frac{5364435496}{256977029}a^{2}-\frac{2122721935}{256977029}a+\frac{587420938}{256977029}$, $\frac{49295894}{256977029}a^{17}-\frac{449736098}{256977029}a^{16}+\frac{1944678682}{256977029}a^{15}-\frac{5715829372}{256977029}a^{14}+\frac{13260697426}{256977029}a^{13}-\frac{26769890056}{256977029}a^{12}+\frac{50085630411}{256977029}a^{11}-\frac{87305487028}{256977029}a^{10}+\frac{136332842052}{256977029}a^{9}-\frac{181687890456}{256977029}a^{8}+\frac{199240150118}{256977029}a^{7}-\frac{174792523862}{256977029}a^{6}+\frac{120213518801}{256977029}a^{5}-\frac{65594260192}{256977029}a^{4}+\frac{30888908408}{256977029}a^{3}-\frac{13681730110}{256977029}a^{2}+\frac{4677291064}{256977029}a-\frac{1156352672}{256977029}$, $\frac{113874829}{256977029}a^{17}-\frac{722195822}{256977029}a^{16}+\frac{2644296712}{256977029}a^{15}-\frac{6983376475}{256977029}a^{14}+\frac{15210951544}{256977029}a^{13}-\frac{29744321256}{256977029}a^{12}+\frac{54347139747}{256977029}a^{11}-\frac{91330094915}{256977029}a^{10}+\frac{134684958419}{256977029}a^{9}-\frac{167079957271}{256977029}a^{8}+\frac{168709550331}{256977029}a^{7}-\frac{135120578443}{256977029}a^{6}+\frac{84468648709}{256977029}a^{5}-\frac{42070936740}{256977029}a^{4}+\frac{18332365019}{256977029}a^{3}-\frac{7366191976}{256977029}a^{2}+\frac{2321635321}{256977029}a-\frac{482848829}{256977029}$, $\frac{202492496}{256977029}a^{17}-\frac{1107849557}{256977029}a^{16}+\frac{3546681864}{256977029}a^{15}-\frac{8408018265}{256977029}a^{14}+\frac{16925029610}{256977029}a^{13}-\frac{31709666108}{256977029}a^{12}+\frac{56220370096}{256977029}a^{11}-\frac{89441144492}{256977029}a^{10}+\frac{119408631927}{256977029}a^{9}-\frac{127351160943}{256977029}a^{8}+\frac{103542410163}{256977029}a^{7}-\frac{60887188397}{256977029}a^{6}+\frac{25308628343}{256977029}a^{5}-\frac{9428100060}{256977029}a^{4}+\frac{4356836593}{256977029}a^{3}-\frac{1035760284}{256977029}a^{2}-\frac{171676903}{256977029}a+\frac{145415210}{256977029}$, $\frac{248398621}{256977029}a^{17}-\frac{1539870317}{256977029}a^{16}+\frac{5357664388}{256977029}a^{15}-\frac{13681203764}{256977029}a^{14}+\frac{29039270452}{256977029}a^{13}-\frac{56135313404}{256977029}a^{12}+\frac{101851555152}{256977029}a^{11}-\frac{168770934096}{256977029}a^{10}+\frac{242458133079}{256977029}a^{9}-\frac{290237520480}{256977029}a^{8}+\frac{281149367423}{256977029}a^{7}-\frac{216027776446}{256977029}a^{6}+\frac{131660415150}{256977029}a^{5}-\frac{66469252389}{256977029}a^{4}+\frac{29816832979}{256977029}a^{3}-\frac{11082255979}{256977029}a^{2}+\frac{3406420124}{256977029}a-\frac{538181094}{256977029}$, $\frac{227808013}{256977029}a^{17}-\frac{1146432692}{256977029}a^{16}+\frac{3491553870}{256977029}a^{15}-\frac{7973971822}{256977029}a^{14}+\frac{15711664610}{256977029}a^{13}-\frac{29189990727}{256977029}a^{12}+\frac{51230832596}{256977029}a^{11}-\frac{79533815662}{256977029}a^{10}+\frac{101840958303}{256977029}a^{9}-\frac{102480598625}{256977029}a^{8}+\frac{76453082427}{256977029}a^{7}-\frac{39137718818}{256977029}a^{6}+\frac{12456026696}{256977029}a^{5}-\frac{2733122133}{256977029}a^{4}-\frac{86436477}{256977029}a^{3}+\frac{1076788047}{256977029}a^{2}-\frac{196969942}{256977029}a+\frac{295630888}{256977029}$, $\frac{258736970}{256977029}a^{17}-\frac{1600769858}{256977029}a^{16}+\frac{5615288854}{256977029}a^{15}-\frac{14410758426}{256977029}a^{14}+\frac{30762834371}{256977029}a^{13}-\frac{59577699281}{256977029}a^{12}+\frac{108243758099}{256977029}a^{11}-\frac{179722602732}{256977029}a^{10}+\frac{259476501062}{256977029}a^{9}-\frac{312895746189}{256977029}a^{8}+\frac{305724619294}{256977029}a^{7}-\frac{236713712462}{256977029}a^{6}+\frac{144190505903}{256977029}a^{5}-\frac{71686204626}{256977029}a^{4}+\frac{31361963896}{256977029}a^{3}-\frac{12110776946}{256977029}a^{2}+\frac{3779050371}{256977029}a-\frac{359690636}{256977029}$
| 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: | \( 476.20211267413316 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 476.20211267413316 \cdot 1}{4\cdot\sqrt{56693912375296000000}}\cr\approx \mathstrut & 0.241313671675920 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 54*x^15 + 116*x^14 - 226*x^13 + 412*x^12 - 686*x^11 + 997*x^10 - 1220*x^9 + 1225*x^8 - 994*x^7 + 652*x^6 - 356*x^5 + 169*x^4 - 70*x^3 + 25*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 - 6*x^17 + 21*x^16 - 54*x^15 + 116*x^14 - 226*x^13 + 412*x^12 - 686*x^11 + 997*x^10 - 1220*x^9 + 1225*x^8 - 994*x^7 + 652*x^6 - 356*x^5 + 169*x^4 - 70*x^3 + 25*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 - 6*x^17 + 21*x^16 - 54*x^15 + 116*x^14 - 226*x^13 + 412*x^12 - 686*x^11 + 997*x^10 - 1220*x^9 + 1225*x^8 - 994*x^7 + 652*x^6 - 356*x^5 + 169*x^4 - 70*x^3 + 25*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 - 6*x^17 + 21*x^16 - 54*x^15 + 116*x^14 - 226*x^13 + 412*x^12 - 686*x^11 + 997*x^10 - 1220*x^9 + 1225*x^8 - 994*x^7 + 652*x^6 - 356*x^5 + 169*x^4 - 70*x^3 + 25*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
$C_6\times S_3$ (as 18T6):
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 36 |
| The 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 3.1.980.1, \(\Q(\zeta_{7})^+\), 6.0.3841600.1, 6.0.153664.1, 9.3.941192000.1 |
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
| Galois closure: | data not computed |
| Degree 12 sibling: | 12.0.153664000000.1 |
| Degree 18 sibling: | 18.6.110730297608000000000.1 |
| Minimal sibling: | 12.0.153664000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
|
\(5\)
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(7\)
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |