Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^15 + x^12 - 58*x^9 + 1639*x^6 - 7062*x^3 + 11323)
gp: K = bnfinit(y^18 - 5*y^15 + y^12 - 58*y^9 + 1639*y^6 - 7062*y^3 + 11323, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 5*x^15 + x^12 - 58*x^9 + 1639*x^6 - 7062*x^3 + 11323);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 5*x^15 + x^12 - 58*x^9 + 1639*x^6 - 7062*x^3 + 11323)
\( x^{18} - 5x^{15} + x^{12} - 58x^{9} + 1639x^{6} - 7062x^{3} + 11323 \)
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: |
\(-272164698131233743036984627\)
\(\medspace = -\,3^{27}\cdot 13^{2}\cdot 19^{6}\cdot 67^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.42\) | 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: | not computed | ||
| Ramified primes: |
\(3\), \(13\), \(19\), \(67\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13}a^{14}-\frac{3}{13}a^{11}-\frac{5}{13}a^{8}-\frac{3}{13}a^{5}-\frac{5}{13}a^{2}$, $\frac{1}{17877527603}a^{15}+\frac{7514677180}{17877527603}a^{12}+\frac{2982387533}{17877527603}a^{9}-\frac{2784275614}{17877527603}a^{6}-\frac{3881024321}{17877527603}a^{3}+\frac{16489939}{105784187}$, $\frac{1}{17877527603}a^{16}+\frac{7514677180}{17877527603}a^{13}+\frac{2982387533}{17877527603}a^{10}-\frac{2784275614}{17877527603}a^{7}-\frac{3881024321}{17877527603}a^{4}+\frac{16489939}{105784187}a$, $\frac{1}{17877527603}a^{17}+\frac{638705025}{17877527603}a^{14}+\frac{5732776395}{17877527603}a^{11}-\frac{4159470045}{17877527603}a^{8}-\frac{1130635459}{17877527603}a^{5}+\frac{108585020}{1375194431}a^{2}$
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 $1$ (assuming GRH)
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: |
\( \frac{1256}{1734167} a^{15} - \frac{3631}{1734167} a^{12} - \frac{16067}{1734167} a^{9} - \frac{99831}{1734167} a^{6} + \frac{1738957}{1734167} a^{3} - \frac{3185067}{1734167} \)
(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: |
$\frac{1389956}{17877527603}a^{15}-\frac{10341891}{17877527603}a^{12}-\frac{22182283}{17877527603}a^{9}-\frac{685001162}{17877527603}a^{6}+\frac{1525447359}{17877527603}a^{3}-\frac{75928793}{105784187}$, $\frac{6135655}{17877527603}a^{15}-\frac{12326722}{17877527603}a^{12}+\frac{50186405}{17877527603}a^{9}-\frac{398125239}{17877527603}a^{6}+\frac{8079071906}{17877527603}a^{3}-\frac{74275983}{105784187}$, $\frac{63842}{1375194431}a^{17}-\frac{20983476}{17877527603}a^{16}-\frac{12605376}{17877527603}a^{15}-\frac{1220092}{1375194431}a^{14}+\frac{55751834}{17877527603}a^{13}+\frac{58908030}{17877527603}a^{12}-\frac{8004636}{1375194431}a^{11}+\frac{233780088}{17877527603}a^{10}+\frac{239343202}{17877527603}a^{9}-\frac{5612847}{1375194431}a^{8}+\frac{2439622661}{17877527603}a^{7}+\frac{607388118}{17877527603}a^{6}+\frac{128378533}{1375194431}a^{5}-\frac{31407188089}{17877527603}a^{4}-\frac{29821676995}{17877527603}a^{3}-\frac{702069429}{1375194431}a^{2}+\frac{451335052}{105784187}a+\frac{343696391}{105784187}$, $\frac{63842}{1375194431}a^{17}+\frac{7525611}{17877527603}a^{16}-\frac{6640087}{17877527603}a^{15}-\frac{1220092}{1375194431}a^{14}-\frac{22668613}{17877527603}a^{13}+\frac{60839464}{17877527603}a^{12}-\frac{8004636}{1375194431}a^{11}+\frac{28004122}{17877527603}a^{10}+\frac{67087392}{17877527603}a^{9}-\frac{5612847}{1375194431}a^{8}-\frac{1083126401}{17877527603}a^{7}+\frac{1668627204}{17877527603}a^{6}+\frac{128378533}{1375194431}a^{5}+\frac{9604519265}{17877527603}a^{4}-\frac{9634002955}{17877527603}a^{3}-\frac{702069429}{1375194431}a^{2}-\frac{255988963}{105784187}a+\frac{360911080}{105784187}$, $\frac{6399853}{17877527603}a^{17}+\frac{4030541}{17877527603}a^{16}+\frac{706496}{17877527603}a^{15}-\frac{29982058}{17877527603}a^{14}-\frac{54413838}{17877527603}a^{13}-\frac{3301630}{17877527603}a^{12}-\frac{155916838}{17877527603}a^{11}+\frac{199719389}{17877527603}a^{10}-\frac{540775212}{17877527603}a^{9}+\frac{680710106}{17877527603}a^{8}-\frac{1757989205}{17877527603}a^{7}+\frac{2655497149}{17877527603}a^{6}+\frac{7935210258}{17877527603}a^{5}+\frac{12352033706}{17877527603}a^{4}+\frac{1882365703}{17877527603}a^{3}-\frac{2855614276}{1375194431}a^{2}-\frac{300544166}{105784187}a-\frac{253923127}{105784187}$, $\frac{8079883}{17877527603}a^{17}+\frac{8717495}{17877527603}a^{16}+\frac{31382775}{17877527603}a^{15}-\frac{13424143}{17877527603}a^{14}+\frac{21763110}{17877527603}a^{13}-\frac{114927603}{17877527603}a^{12}+\frac{89717117}{17877527603}a^{11}-\frac{313028608}{17877527603}a^{10}+\frac{81332744}{17877527603}a^{9}-\frac{1155392347}{17877527603}a^{8}-\frac{700208699}{17877527603}a^{7}-\frac{1832088035}{17877527603}a^{6}+\frac{7504789548}{17877527603}a^{5}+\frac{17569874339}{17877527603}a^{4}+\frac{29955290247}{17877527603}a^{3}+\frac{414954377}{1375194431}a^{2}-\frac{122989178}{105784187}a-\frac{217202381}{105784187}$, $\frac{20376141}{17877527603}a^{17}+\frac{1584695}{17877527603}a^{16}-\frac{55513264}{17877527603}a^{15}-\frac{2985884}{17877527603}a^{14}-\frac{67984642}{17877527603}a^{13}+\frac{213271763}{17877527603}a^{12}-\frac{100426044}{17877527603}a^{11}-\frac{95632057}{17877527603}a^{10}+\frac{418995400}{17877527603}a^{9}-\frac{1613756656}{17877527603}a^{8}-\frac{199124521}{17877527603}a^{7}+\frac{4118241715}{17877527603}a^{6}+\frac{25932865331}{17877527603}a^{5}+\frac{5087144568}{17877527603}a^{4}-\frac{86066430718}{17877527603}a^{3}-\frac{929681982}{1375194431}a^{2}-\frac{449615192}{105784187}a+\frac{1693672751}{105784187}$, $\frac{20376141}{17877527603}a^{17}-\frac{30293850}{17877527603}a^{16}-\frac{32319179}{17877527603}a^{15}-\frac{2985884}{17877527603}a^{14}+\frac{151186340}{17877527603}a^{13}-\frac{15316287}{17877527603}a^{12}-\frac{100426044}{17877527603}a^{11}+\frac{353733507}{17877527603}a^{10}+\frac{400190966}{17877527603}a^{9}-\frac{1613756656}{17877527603}a^{8}+\frac{2292833458}{17877527603}a^{7}+\frac{3661101380}{17877527603}a^{6}+\frac{25932865331}{17877527603}a^{5}-\frac{51981379193}{17877527603}a^{4}-\frac{38578256815}{17877527603}a^{3}-\frac{929681982}{1375194431}a^{2}+\frac{927694950}{105784187}a-\frac{238781520}{105784187}$
| 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: | \( 1639391.44819 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 1639391.44819 \cdot 1}{6\cdot\sqrt{272164698131233743036984627}}\cr\approx \mathstrut & 0.252775124833 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^15 + x^12 - 58*x^9 + 1639*x^6 - 7062*x^3 + 11323)
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 - 5*x^15 + x^12 - 58*x^9 + 1639*x^6 - 7062*x^3 + 11323, 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 - 5*x^15 + x^12 - 58*x^9 + 1639*x^6 - 7062*x^3 + 11323);
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 - 5*x^15 + x^12 - 58*x^9 + 1639*x^6 - 7062*x^3 + 11323);
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
$C_3\wr (C_3\times S_3)$ (as 18T584):
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 13122 |
| The 170 conjugacy class representatives for $C_3\wr (C_3\times S_3)$ |
| Character table for $C_3\wr (C_3\times S_3)$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 6.0.9747.1 |
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 |
| 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 | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | $18$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | R | $18$ | $18$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $6$ | $3$ | $27$ | |||
|
\(13\)
| 13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 19.3.2.2 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
|
\(67\)
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 67.3.2.2 | $x^{3} + 268$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 67.9.0.1 | $x^{9} + 25 x^{3} + 49 x^{2} + 55 x + 65$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |