Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 7*x^15 + 5*x^14 + 6*x^13 + 21*x^12 - 29*x^11 - 13*x^10 - 11*x^9 + 49*x^8 - 8*x^7 - 3*x^6 - 24*x^5 + 14*x^4 + 3*x^3 - 3*x + 1)
gp: K = bnfinit(y^18 - y^17 - 7*y^15 + 5*y^14 + 6*y^13 + 21*y^12 - 29*y^11 - 13*y^10 - 11*y^9 + 49*y^8 - 8*y^7 - 3*y^6 - 24*y^5 + 14*y^4 + 3*y^3 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 - 7*x^15 + 5*x^14 + 6*x^13 + 21*x^12 - 29*x^11 - 13*x^10 - 11*x^9 + 49*x^8 - 8*x^7 - 3*x^6 - 24*x^5 + 14*x^4 + 3*x^3 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 - 7*x^15 + 5*x^14 + 6*x^13 + 21*x^12 - 29*x^11 - 13*x^10 - 11*x^9 + 49*x^8 - 8*x^7 - 3*x^6 - 24*x^5 + 14*x^4 + 3*x^3 - 3*x + 1)
\( x^{18} - x^{17} - 7 x^{15} + 5 x^{14} + 6 x^{13} + 21 x^{12} - 29 x^{11} - 13 x^{10} - 11 x^{9} + \cdots + 1 \)
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: |
\(-1963375693486812487\)
\(\medspace = -\,31^{6}\cdot 1303^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.38\) | 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: | $31^{1/2}1303^{1/2}\approx 200.98009851724126$ | ||
| Ramified primes: |
\(31\), \(1303\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1303}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{56280053}a^{17}+\frac{5628454}{56280053}a^{16}+\frac{21025400}{56280053}a^{15}+\frac{73469}{56280053}a^{14}+\frac{27411009}{56280053}a^{13}-\frac{4228859}{56280053}a^{12}+\frac{17431936}{56280053}a^{11}-\frac{6297798}{56280053}a^{10}-\frac{6861113}{56280053}a^{9}-\frac{6923628}{56280053}a^{8}-\frac{6896537}{56280053}a^{7}+\frac{10914234}{56280053}a^{6}-\frac{22001616}{56280053}a^{5}-\frac{22605443}{56280053}a^{4}+\frac{4137874}{56280053}a^{3}+\frac{26072213}{56280053}a^{2}-\frac{20972875}{56280053}a-\frac{4593013}{56280053}$
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: |
$\frac{278609983}{56280053}a^{17}-\frac{81374431}{56280053}a^{16}-\frac{73955229}{56280053}a^{15}-\frac{1997077087}{56280053}a^{14}-\frac{21827697}{56280053}a^{13}+\frac{1777917901}{56280053}a^{12}+\frac{7109845976}{56280053}a^{11}-\frac{3121632387}{56280053}a^{10}-\frac{6254930709}{56280053}a^{9}-\frac{7357936786}{56280053}a^{8}+\frac{8730739889}{56280053}a^{7}+\frac{4467489135}{56280053}a^{6}+\frac{1950844743}{56280053}a^{5}-\frac{5599019806}{56280053}a^{4}-\frac{263867783}{56280053}a^{3}+\frac{908515748}{56280053}a^{2}+\frac{743011827}{56280053}a-\frac{404729600}{56280053}$, $\frac{72971847}{56280053}a^{17}+\frac{26872940}{56280053}a^{16}-\frac{15450277}{56280053}a^{15}-\frac{519461961}{56280053}a^{14}-\frac{355953318}{56280053}a^{13}+\frac{324790614}{56280053}a^{12}+\frac{2016762813}{56280053}a^{11}+\frac{474765802}{56280053}a^{10}-\frac{1580798506}{56280053}a^{9}-\frac{2649158750}{56280053}a^{8}+\frac{545969670}{56280053}a^{7}+\frac{1696396757}{56280053}a^{6}+\frac{1220985029}{56280053}a^{5}-\frac{453740641}{56280053}a^{4}-\frac{479277658}{56280053}a^{3}+\frac{82771580}{56280053}a^{2}+\frac{133824599}{56280053}a-\frac{38157874}{56280053}$, $\frac{119784484}{56280053}a^{17}-\frac{90200729}{56280053}a^{16}-\frac{36001666}{56280053}a^{15}-\frac{853553356}{56280053}a^{14}+\frac{383331496}{56280053}a^{13}+\frac{919686507}{56280053}a^{12}+\frac{2817183167}{56280053}a^{11}-\frac{2791996246}{56280053}a^{10}-\frac{2669296780}{56280053}a^{9}-\frac{2147341648}{56280053}a^{8}+\frac{5536625677}{56280053}a^{7}+\frac{1154504353}{56280053}a^{6}-\frac{63950552}{56280053}a^{5}-\frac{3279500149}{56280053}a^{4}+\frac{359461369}{56280053}a^{3}+\frac{592311082}{56280053}a^{2}+\frac{281459002}{56280053}a-\frac{237263333}{56280053}$, $\frac{254348217}{56280053}a^{17}-\frac{83066585}{56280053}a^{16}-\frac{69508162}{56280053}a^{15}-\frac{1814084566}{56280053}a^{14}+\frac{41630845}{56280053}a^{13}+\frac{1646922782}{56280053}a^{12}+\frac{6397567160}{56280053}a^{11}-\frac{3092891720}{56280053}a^{10}-\frac{5638465689}{56280053}a^{9}-\frac{6314358202}{56280053}a^{8}+\frac{8161433217}{56280053}a^{7}+\frac{3638036997}{56280053}a^{6}+\frac{1379090241}{56280053}a^{5}-\frac{4889870719}{56280053}a^{4}+\frac{76508239}{56280053}a^{3}+\frac{888619447}{56280053}a^{2}+\frac{457145424}{56280053}a-\frac{359308695}{56280053}$, $\frac{129308393}{56280053}a^{17}-\frac{43454741}{56280053}a^{16}-\frac{18046642}{56280053}a^{15}-\frac{927662037}{56280053}a^{14}+\frac{22480195}{56280053}a^{13}+\frac{726644483}{56280053}a^{12}+\frac{3248917033}{56280053}a^{11}-\frac{1474043401}{56280053}a^{10}-\frac{2485411052}{56280053}a^{9}-\frac{3427227746}{56280053}a^{8}+\frac{3774674869}{56280053}a^{7}+\frac{1613677398}{56280053}a^{6}+\frac{1228662846}{56280053}a^{5}-\frac{2309591166}{56280053}a^{4}-\frac{55461514}{56280053}a^{3}+\frac{243873685}{56280053}a^{2}+\frac{233433086}{56280053}a-\frac{60196953}{56280053}$, $\frac{77400539}{56280053}a^{17}-\frac{798857}{56280053}a^{16}-\frac{4681659}{56280053}a^{15}-\frac{559155859}{56280053}a^{14}-\frac{163282818}{56280053}a^{13}+\frac{324004996}{56280053}a^{12}+\frac{2049129206}{56280053}a^{11}-\frac{220949575}{56280053}a^{10}-\frac{1342477631}{56280053}a^{9}-\frac{2485217601}{56280053}a^{8}+\frac{1430173484}{56280053}a^{7}+\frac{1015999317}{56280053}a^{6}+\frac{1115058240}{56280053}a^{5}-\frac{914433737}{56280053}a^{4}-\frac{7613756}{56280053}a^{3}+\frac{38770586}{56280053}a^{2}+\frac{78364179}{56280053}a-\frac{59931345}{56280053}$, $\frac{10395034}{56280053}a^{17}-\frac{40760675}{56280053}a^{16}-\frac{10918296}{56280053}a^{15}-\frac{63846317}{56280053}a^{14}+\frac{285117673}{56280053}a^{13}+\frac{203271140}{56280053}a^{12}+\frac{61681141}{56280053}a^{11}-\frac{1260645903}{56280053}a^{10}-\frac{324348327}{56280053}a^{9}+\frac{708914108}{56280053}a^{8}+\frac{1989836209}{56280053}a^{7}-\frac{506248161}{56280053}a^{6}-\frac{1113117731}{56280053}a^{5}-\frac{1057281812}{56280053}a^{4}+\frac{691331883}{56280053}a^{3}+\frac{442964502}{56280053}a^{2}+\frac{37284993}{56280053}a-\frac{187535687}{56280053}$, $\frac{74387275}{56280053}a^{17}+\frac{31638890}{56280053}a^{16}-\frac{10844576}{56280053}a^{15}-\frac{534920173}{56280053}a^{14}-\frac{389243606}{56280053}a^{13}+\frac{288110724}{56280053}a^{12}+\frac{2097875850}{56280053}a^{11}+\frac{592732928}{56280053}a^{10}-\frac{1531424402}{56280053}a^{9}-\frac{2894423015}{56280053}a^{8}+\frac{488189419}{56280053}a^{7}+\frac{1753330992}{56280053}a^{6}+\frac{1409620295}{56280053}a^{5}-\frac{607543791}{56280053}a^{4}-\frac{456553084}{56280053}a^{3}+\frac{85801167}{56280053}a^{2}+\frac{184904638}{56280053}a-\frac{39600249}{56280053}$
| 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: | \( 29.661621596 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 29.661621596 \cdot 1}{2\cdot\sqrt{1963375693486812487}}\cr\approx \mathstrut & 0.16154071643 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 7*x^15 + 5*x^14 + 6*x^13 + 21*x^12 - 29*x^11 - 13*x^10 - 11*x^9 + 49*x^8 - 8*x^7 - 3*x^6 - 24*x^5 + 14*x^4 + 3*x^3 - 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 - x^17 - 7*x^15 + 5*x^14 + 6*x^13 + 21*x^12 - 29*x^11 - 13*x^10 - 11*x^9 + 49*x^8 - 8*x^7 - 3*x^6 - 24*x^5 + 14*x^4 + 3*x^3 - 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 - x^17 - 7*x^15 + 5*x^14 + 6*x^13 + 21*x^12 - 29*x^11 - 13*x^10 - 11*x^9 + 49*x^8 - 8*x^7 - 3*x^6 - 24*x^5 + 14*x^4 + 3*x^3 - 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 - x^17 - 7*x^15 + 5*x^14 + 6*x^13 + 21*x^12 - 29*x^11 - 13*x^10 - 11*x^9 + 49*x^8 - 8*x^7 - 3*x^6 - 24*x^5 + 14*x^4 + 3*x^3 - 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
$S_3\wr S_3$ (as 18T314):
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 1296 |
| The 22 conjugacy class representatives for $S_3\wr S_3$ |
| Character table for $S_3\wr S_3$ |
Intermediate fields
| 3.1.31.1, 6.0.1252183.1, 9.1.38817673.1 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 sibling: | data not computed |
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 27 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 9.1.38817673.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(1303\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |