Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 41*x^16 - 124*x^15 + 276*x^14 - 476*x^13 + 646*x^12 - 678*x^11 + 517*x^10 - 231*x^9 - 15*x^8 + 98*x^7 - 48*x^6 - 18*x^5 + 36*x^4 - 22*x^3 + 9*x^2 - 3*x + 1)
gp: K = bnfinit(y^18 - 9*y^17 + 41*y^16 - 124*y^15 + 276*y^14 - 476*y^13 + 646*y^12 - 678*y^11 + 517*y^10 - 231*y^9 - 15*y^8 + 98*y^7 - 48*y^6 - 18*y^5 + 36*y^4 - 22*y^3 + 9*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 41*x^16 - 124*x^15 + 276*x^14 - 476*x^13 + 646*x^12 - 678*x^11 + 517*x^10 - 231*x^9 - 15*x^8 + 98*x^7 - 48*x^6 - 18*x^5 + 36*x^4 - 22*x^3 + 9*x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 41*x^16 - 124*x^15 + 276*x^14 - 476*x^13 + 646*x^12 - 678*x^11 + 517*x^10 - 231*x^9 - 15*x^8 + 98*x^7 - 48*x^6 - 18*x^5 + 36*x^4 - 22*x^3 + 9*x^2 - 3*x + 1)
\( x^{18} - 9 x^{17} + 41 x^{16} - 124 x^{15} + 276 x^{14} - 476 x^{13} + 646 x^{12} - 678 x^{11} + \cdots + 1 \)
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: |
\(-12906538539418320896\)
\(\medspace = -\,2^{16}\cdot 11^{9}\cdot 17^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.53\) | 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: | $2^{8/9}11^{1/2}17^{2/3}\approx 40.60484358442798$ | ||
| Ramified primes: |
\(2\), \(11\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{401}a^{16}-\frac{8}{401}a^{15}+\frac{20}{401}a^{14}+\frac{16}{401}a^{12}-\frac{59}{401}a^{11}-\frac{22}{401}a^{10}+\frac{67}{401}a^{9}+\frac{68}{401}a^{8}+\frac{169}{401}a^{7}+\frac{72}{401}a^{6}-\frac{22}{401}a^{5}+\frac{197}{401}a^{4}+\frac{64}{401}a^{3}-\frac{55}{401}a^{2}-\frac{107}{401}a-\frac{185}{401}$, $\frac{1}{42907}a^{17}+\frac{45}{42907}a^{16}-\frac{20855}{42907}a^{15}+\frac{4268}{42907}a^{14}-\frac{21237}{42907}a^{13}+\frac{11215}{42907}a^{12}+\frac{18505}{42907}a^{11}+\frac{14941}{42907}a^{10}-\frac{9614}{42907}a^{9}+\frac{20214}{42907}a^{8}+\frac{20257}{42907}a^{7}-\frac{16256}{42907}a^{6}+\frac{17076}{42907}a^{5}+\frac{19327}{42907}a^{4}+\frac{9753}{42907}a^{3}+\frac{20637}{42907}a^{2}-\frac{5455}{42907}a+\frac{13052}{42907}$
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: |
$\frac{44552}{42907}a^{17}-\frac{347127}{42907}a^{16}+\frac{1371712}{42907}a^{15}-\frac{3590437}{42907}a^{14}+\frac{6856553}{42907}a^{13}-\frac{10000699}{42907}a^{12}+\frac{11008669}{42907}a^{11}-\frac{8377199}{42907}a^{10}+\frac{3122579}{42907}a^{9}+\frac{1567297}{42907}a^{8}-\frac{2711090}{42907}a^{7}+\frac{817479}{42907}a^{6}+\frac{798600}{42907}a^{5}-\frac{844118}{42907}a^{4}+\frac{288684}{42907}a^{3}+\frac{44915}{42907}a^{2}+\frac{5002}{42907}a+\frac{11048}{42907}$, $\frac{103056}{42907}a^{17}-\frac{785240}{42907}a^{16}+\frac{3028250}{42907}a^{15}-\frac{7706213}{42907}a^{14}+\frac{14245108}{42907}a^{13}-\frac{20013687}{42907}a^{12}+\frac{20973622}{42907}a^{11}-\frac{14655840}{42907}a^{10}+\frac{4072874}{42907}a^{9}+\frac{3981656}{42907}a^{8}-\frac{4492848}{42907}a^{7}+\frac{212087}{42907}a^{6}+\frac{1860819}{42907}a^{5}-\frac{865909}{42907}a^{4}+\frac{155497}{42907}a^{3}-\frac{128075}{42907}a^{2}+\frac{172621}{42907}a+\frac{40309}{42907}$, $\frac{16775}{42907}a^{17}-\frac{99841}{42907}a^{16}+\frac{251103}{42907}a^{15}-\frac{205045}{42907}a^{14}-\frac{680366}{42907}a^{13}+\frac{2999797}{42907}a^{12}-\frac{6605929}{42907}a^{11}+\frac{10023305}{42907}a^{10}-\frac{10742221}{42907}a^{9}+\frac{7437037}{42907}a^{8}-\frac{1793494}{42907}a^{7}-\frac{2219586}{42907}a^{6}+\frac{2201511}{42907}a^{5}-\frac{221386}{42907}a^{4}-\frac{722634}{42907}a^{3}+\frac{596005}{42907}a^{2}-\frac{224641}{42907}a+\frac{88858}{42907}$, $\frac{126460}{42907}a^{17}-\frac{1099092}{42907}a^{16}+\frac{4802020}{42907}a^{15}-\frac{13850019}{42907}a^{14}+\frac{29223591}{42907}a^{13}-\frac{47450368}{42907}a^{12}+\frac{59867761}{42907}a^{11}-\frac{56720689}{42907}a^{10}+\frac{36180863}{42907}a^{9}-\frac{9044905}{42907}a^{8}-\frac{8141678}{42907}a^{7}+\frac{8367716}{42907}a^{6}-\frac{748705}{42907}a^{5}-\frac{3243252}{42907}a^{4}+\frac{2635400}{42907}a^{3}-\frac{1159898}{42907}a^{2}+\frac{497843}{42907}a-\frac{153089}{42907}$, $\frac{113472}{42907}a^{17}-\frac{893143}{42907}a^{16}+\frac{3534623}{42907}a^{15}-\frac{9194599}{42907}a^{14}+\frac{17315405}{42907}a^{13}-\frac{24668927}{42907}a^{12}+\frac{26127804}{42907}a^{11}-\frac{18273027}{42907}a^{10}+\frac{4490017}{42907}a^{9}+\frac{6352550}{42907}a^{8}-\frac{7180286}{42907}a^{7}+\frac{775524}{42907}a^{6}+\frac{3190271}{42907}a^{5}-\frac{2011005}{42907}a^{4}+\frac{220931}{42907}a^{3}+\frac{169188}{42907}a^{2}+\frac{74790}{42907}a+\frac{26111}{42907}$, $\frac{128105}{42907}a^{17}-\frac{1117729}{42907}a^{16}+\frac{4875929}{42907}a^{15}-\frac{14002867}{42907}a^{14}+\frac{29343745}{42907}a^{13}-\frac{47175108}{42907}a^{12}+\frac{58660989}{42907}a^{11}-\frac{54175007}{42907}a^{10}+\frac{32564582}{42907}a^{9}-\frac{5478206}{42907}a^{8}-\frac{10344732}{42907}a^{7}+\frac{8636927}{42907}a^{6}+\frac{31491}{42907}a^{5}-\frac{3864061}{42907}a^{4}+\frac{2665572}{42907}a^{3}-\frac{946458}{42907}a^{2}+\frac{366527}{42907}a-\frac{113472}{42907}$, $\frac{3808}{42907}a^{17}-\frac{55694}{42907}a^{16}+\frac{319704}{42907}a^{15}-\frac{1117729}{42907}a^{14}+\frac{2755247}{42907}a^{13}-\frac{5120354}{42907}a^{12}+\frac{7402952}{42907}a^{11}-\frac{8219522}{42907}a^{10}+\frac{6573697}{42907}a^{9}-\frac{3039795}{42907}a^{8}-\frac{364654}{42907}a^{7}+\frac{1599191}{42907}a^{6}-\frac{818861}{42907}a^{5}-\frac{223368}{42907}a^{4}+\frac{510863}{42907}a^{3}-\frac{232430}{42907}a^{2}+\frac{46671}{42907}a-\frac{71067}{42907}$, $\frac{3808}{42907}a^{17}-\frac{9042}{42907}a^{16}-\frac{53512}{42907}a^{15}+\frac{416009}{42907}a^{14}-\frac{1449639}{42907}a^{13}+\frac{3349338}{42907}a^{12}-\frac{5733010}{42907}a^{11}+\frac{7444957}{42907}a^{10}-\frac{7077256}{42907}a^{9}+\frac{4294520}{42907}a^{8}-\frac{546982}{42907}a^{7}-\frac{1692450}{42907}a^{6}+\frac{1372820}{42907}a^{5}+\frac{42420}{42907}a^{4}-\frac{708295}{42907}a^{3}+\frac{419735}{42907}a^{2}-\frac{53695}{42907}a+\frac{8434}{42907}$
| 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: | \( 113.075873428 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 113.075873428 \cdot 1}{2\cdot\sqrt{12906538539418320896}}\cr\approx \mathstrut & 0.240189423250 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 41*x^16 - 124*x^15 + 276*x^14 - 476*x^13 + 646*x^12 - 678*x^11 + 517*x^10 - 231*x^9 - 15*x^8 + 98*x^7 - 48*x^6 - 18*x^5 + 36*x^4 - 22*x^3 + 9*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 - 9*x^17 + 41*x^16 - 124*x^15 + 276*x^14 - 476*x^13 + 646*x^12 - 678*x^11 + 517*x^10 - 231*x^9 - 15*x^8 + 98*x^7 - 48*x^6 - 18*x^5 + 36*x^4 - 22*x^3 + 9*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 - 9*x^17 + 41*x^16 - 124*x^15 + 276*x^14 - 476*x^13 + 646*x^12 - 678*x^11 + 517*x^10 - 231*x^9 - 15*x^8 + 98*x^7 - 48*x^6 - 18*x^5 + 36*x^4 - 22*x^3 + 9*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 - 9*x^17 + 41*x^16 - 124*x^15 + 276*x^14 - 476*x^13 + 646*x^12 - 678*x^11 + 517*x^10 - 231*x^9 - 15*x^8 + 98*x^7 - 48*x^6 - 18*x^5 + 36*x^4 - 22*x^3 + 9*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
| \(\Q(\sqrt{-11}) \), 3.1.44.1 x3, 6.0.21296.1, 9.1.1083199744.1 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: | 9.1.1083199744.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.16.1 | $x^{18} + 9 x^{17} + 45 x^{16} + 156 x^{15} + 414 x^{14} + 882 x^{13} + 1554 x^{12} + 2304 x^{11} + 2911 x^{10} + 3163 x^{9} + 2889 x^{8} + 1968 x^{7} + 798 x^{6} + 378 x^{5} + 750 x^{4} + 732 x^{3} + 247 x^{2} + 17 x + 7$ | $9$ | $2$ | $16$ | 18T18 | $[\ ]_{9}^{6}$ |
|
\(11\)
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(17\)
| 17.6.4.1 | $x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 17.6.0.1 | $x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 17.6.0.1 | $x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |