Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 2*x^16 - 4*x^15 + 6*x^14 - 20*x^13 + 22*x^12 - 27*x^11 + 19*x^10 - 11*x^9 + 16*x^8 - 5*x^7 + 2*x^6 - 5*x^5 - 9*x^4 - 7*x^3 - x^2 + 2*x + 1)
gp: K = bnfinit(y^18 - y^17 + 2*y^16 - 4*y^15 + 6*y^14 - 20*y^13 + 22*y^12 - 27*y^11 + 19*y^10 - 11*y^9 + 16*y^8 - 5*y^7 + 2*y^6 - 5*y^5 - 9*y^4 - 7*y^3 - y^2 + 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 + 2*x^16 - 4*x^15 + 6*x^14 - 20*x^13 + 22*x^12 - 27*x^11 + 19*x^10 - 11*x^9 + 16*x^8 - 5*x^7 + 2*x^6 - 5*x^5 - 9*x^4 - 7*x^3 - x^2 + 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 + 2*x^16 - 4*x^15 + 6*x^14 - 20*x^13 + 22*x^12 - 27*x^11 + 19*x^10 - 11*x^9 + 16*x^8 - 5*x^7 + 2*x^6 - 5*x^5 - 9*x^4 - 7*x^3 - x^2 + 2*x + 1)
\( x^{18} - x^{17} + 2 x^{16} - 4 x^{15} + 6 x^{14} - 20 x^{13} + 22 x^{12} - 27 x^{11} + 19 x^{10} + \cdots + 1 \)
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: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(63147574802143803281\)
\(\medspace = 23^{7}\cdot 2647^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.59\) | 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: | $23^{1/2}2647^{1/2}\approx 246.74075463935827$ | ||
| Ramified primes: |
\(23\), \(2647\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{60881}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{50807029337}a^{17}+\frac{16407459087}{50807029337}a^{16}-\frac{9696592112}{50807029337}a^{15}-\frac{17504046562}{50807029337}a^{14}+\frac{18980972274}{50807029337}a^{13}-\frac{12878990356}{50807029337}a^{12}+\frac{22347333426}{50807029337}a^{11}+\frac{20395200149}{50807029337}a^{10}+\frac{25011516902}{50807029337}a^{9}-\frac{18343599126}{50807029337}a^{8}+\frac{13327671496}{50807029337}a^{7}-\frac{12979758465}{50807029337}a^{6}-\frac{23465587513}{50807029337}a^{5}-\frac{20384985081}{50807029337}a^{4}-\frac{10044890585}{50807029337}a^{3}-\frac{1541833085}{50807029337}a^{2}+\frac{1876203166}{50807029337}a+\frac{20954518826}{50807029337}$
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: | $9$ | 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: |
$\frac{28609942}{179530139}a^{17}+\frac{1788435}{179530139}a^{16}+\frac{32205268}{179530139}a^{15}-\frac{68015741}{179530139}a^{14}+\frac{99042163}{179530139}a^{13}-\frac{475989239}{179530139}a^{12}+\frac{165265217}{179530139}a^{11}-\frac{400774060}{179530139}a^{10}+\frac{240179630}{179530139}a^{9}-\frac{728572731}{179530139}a^{8}+\frac{1387553328}{179530139}a^{7}-\frac{932389371}{179530139}a^{6}+\frac{740260034}{179530139}a^{5}-\frac{439365065}{179530139}a^{4}-\frac{257434978}{179530139}a^{3}-\frac{542372411}{179530139}a^{2}-\frac{250012871}{179530139}a+\frac{86284415}{179530139}$, $\frac{6336154234}{50807029337}a^{17}-\frac{16702359167}{50807029337}a^{16}+\frac{26005354524}{50807029337}a^{15}-\frac{50560342673}{50807029337}a^{14}+\frac{89450521773}{50807029337}a^{13}-\frac{210820721138}{50807029337}a^{12}+\frac{378912887731}{50807029337}a^{11}-\frac{486713109146}{50807029337}a^{10}+\frac{529926760101}{50807029337}a^{9}-\frac{455951566562}{50807029337}a^{8}+\frac{454457752513}{50807029337}a^{7}-\frac{410198857022}{50807029337}a^{6}+\frac{301768568115}{50807029337}a^{5}-\frac{221460582711}{50807029337}a^{4}+\frac{106493456532}{50807029337}a^{3}-\frac{26593343475}{50807029337}a^{2}+\frac{27418409457}{50807029337}a+\frac{18883937936}{50807029337}$, $\frac{2327588013}{50807029337}a^{17}+\frac{879018172}{50807029337}a^{16}+\frac{3175479029}{50807029337}a^{15}-\frac{3352661544}{50807029337}a^{14}+\frac{4333948011}{50807029337}a^{13}-\frac{30442344945}{50807029337}a^{12}-\frac{9379265364}{50807029337}a^{11}-\frac{21530360182}{50807029337}a^{10}-\frac{30292219097}{50807029337}a^{9}+\frac{6123904585}{50807029337}a^{8}-\frac{16142211697}{50807029337}a^{7}+\frac{65749579189}{50807029337}a^{6}+\frac{16237947562}{50807029337}a^{5}-\frac{29808901183}{50807029337}a^{4}+\frac{55527296211}{50807029337}a^{3}-\frac{77404193944}{50807029337}a^{2}-\frac{25050512667}{50807029337}a-\frac{41436276237}{50807029337}$, $\frac{8389319948}{50807029337}a^{17}-\frac{16351372799}{50807029337}a^{16}+\frac{23789709492}{50807029337}a^{15}-\frac{50729774245}{50807029337}a^{14}+\frac{81374128577}{50807029337}a^{13}-\frac{215335253448}{50807029337}a^{12}+\frac{342641470293}{50807029337}a^{11}-\frac{391758882872}{50807029337}a^{10}+\frac{390600002137}{50807029337}a^{9}-\frac{241834029060}{50807029337}a^{8}+\frac{242028534612}{50807029337}a^{7}-\frac{145855110271}{50807029337}a^{6}+\frac{49966049786}{50807029337}a^{5}-\frac{78320256022}{50807029337}a^{4}-\frac{4453929163}{50807029337}a^{3}-\frac{43401710454}{50807029337}a^{2}+\frac{49778559694}{50807029337}a+\frac{34940168168}{50807029337}$, $a$, $\frac{23869413077}{50807029337}a^{17}-\frac{36638974190}{50807029337}a^{16}+\frac{69373738382}{50807029337}a^{15}-\frac{128842473390}{50807029337}a^{14}+\frac{200050996135}{50807029337}a^{13}-\frac{566342922400}{50807029337}a^{12}+\frac{800879093398}{50807029337}a^{11}-\frac{1043120499145}{50807029337}a^{10}+\frac{877725461220}{50807029337}a^{9}-\frac{464902733861}{50807029337}a^{8}+\frac{266461933928}{50807029337}a^{7}+\frac{18444744029}{50807029337}a^{6}-\frac{39387134706}{50807029337}a^{5}-\frac{55774493374}{50807029337}a^{4}-\frac{251361761975}{50807029337}a^{3}-\frac{2873919419}{50807029337}a^{2}-\frac{20384606446}{50807029337}a-\frac{4353490554}{50807029337}$, $\frac{22039474632}{50807029337}a^{17}-\frac{25110115581}{50807029337}a^{16}+\frac{55509757144}{50807029337}a^{15}-\frac{100471076570}{50807029337}a^{14}+\frac{157306364706}{50807029337}a^{13}-\frac{485821354299}{50807029337}a^{12}+\frac{586611833423}{50807029337}a^{11}-\frac{808805799151}{50807029337}a^{10}+\frac{636107514209}{50807029337}a^{9}-\frac{448237963312}{50807029337}a^{8}+\frac{440818979792}{50807029337}a^{7}-\frac{207918755883}{50807029337}a^{6}+\frac{143919158989}{50807029337}a^{5}-\frac{161085925577}{50807029337}a^{4}-\frac{181266427709}{50807029337}a^{3}-\frac{113661039088}{50807029337}a^{2}-\frac{52738347283}{50807029337}a+\frac{9417544630}{50807029337}$, $\frac{3357085508}{50807029337}a^{17}-\frac{6728348366}{50807029337}a^{16}+\frac{16787608688}{50807029337}a^{15}-\frac{33384973999}{50807029337}a^{14}+\frac{57679113676}{50807029337}a^{13}-\frac{123942933203}{50807029337}a^{12}+\frac{205340345738}{50807029337}a^{11}-\frac{335111096038}{50807029337}a^{10}+\frac{425004269720}{50807029337}a^{9}-\frac{448494319941}{50807029337}a^{8}+\frac{320104614423}{50807029337}a^{7}-\frac{159883613237}{50807029337}a^{6}+\frac{3610431648}{50807029337}a^{5}-\frac{28734464308}{50807029337}a^{4}-\frac{50148553644}{50807029337}a^{3}+\frac{37654855329}{50807029337}a^{2}-\frac{48412208293}{50807029337}a-\frac{21690017811}{50807029337}$, $\frac{12068555269}{50807029337}a^{17}-\frac{8232770533}{50807029337}a^{16}+\frac{29051357167}{50807029337}a^{15}-\frac{57007495080}{50807029337}a^{14}+\frac{85618181133}{50807029337}a^{13}-\frac{267400944243}{50807029337}a^{12}+\frac{269798602718}{50807029337}a^{11}-\frac{466970919443}{50807029337}a^{10}+\frac{467355947416}{50807029337}a^{9}-\frac{492535244877}{50807029337}a^{8}+\frac{492017763363}{50807029337}a^{7}-\frac{170428699948}{50807029337}a^{6}+\frac{145171954699}{50807029337}a^{5}-\frac{101962808318}{50807029337}a^{4}-\frac{78204575089}{50807029337}a^{3}-\frac{92148588057}{50807029337}a^{2}-\frac{109466195287}{50807029337}a-\frac{10454095099}{50807029337}$
| 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: | \( 291.939159084 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 291.939159084 \cdot 1}{2\cdot\sqrt{63147574802143803281}}\cr\approx \mathstrut & 0.178477287635 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 2*x^16 - 4*x^15 + 6*x^14 - 20*x^13 + 22*x^12 - 27*x^11 + 19*x^10 - 11*x^9 + 16*x^8 - 5*x^7 + 2*x^6 - 5*x^5 - 9*x^4 - 7*x^3 - x^2 + 2*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 - x^17 + 2*x^16 - 4*x^15 + 6*x^14 - 20*x^13 + 22*x^12 - 27*x^11 + 19*x^10 - 11*x^9 + 16*x^8 - 5*x^7 + 2*x^6 - 5*x^5 - 9*x^4 - 7*x^3 - x^2 + 2*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 - x^17 + 2*x^16 - 4*x^15 + 6*x^14 - 20*x^13 + 22*x^12 - 27*x^11 + 19*x^10 - 11*x^9 + 16*x^8 - 5*x^7 + 2*x^6 - 5*x^5 - 9*x^4 - 7*x^3 - x^2 + 2*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 - x^17 + 2*x^16 - 4*x^15 + 6*x^14 - 20*x^13 + 22*x^12 - 27*x^11 + 19*x^10 - 11*x^9 + 16*x^8 - 5*x^7 + 2*x^6 - 5*x^5 - 9*x^4 - 7*x^3 - x^2 + 2*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
$S_4^3.S_4$ (as 18T880):
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 331776 |
| The 165 conjugacy class representatives for $S_4^3.S_4$ |
| Character table for $S_4^3.S_4$ |
Intermediate fields
| 3.1.23.1, 9.1.32206049.1 |
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 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(2647\)
| $\Q_{2647}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2647}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |