Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 126*x^16 + 6615*x^14 - 187278*x^12 + 3090087*x^10 - 29950074*x^8 + 163061514*x^6 - 444713220*x^4 + 466948881*x^2 - 54059887)
gp: K = bnfinit(y^18 - 126*y^16 + 6615*y^14 - 187278*y^12 + 3090087*y^10 - 29950074*y^8 + 163061514*y^6 - 444713220*y^4 + 466948881*y^2 - 54059887, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 126*x^16 + 6615*x^14 - 187278*x^12 + 3090087*x^10 - 29950074*x^8 + 163061514*x^6 - 444713220*x^4 + 466948881*x^2 - 54059887);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 126*x^16 + 6615*x^14 - 187278*x^12 + 3090087*x^10 - 29950074*x^8 + 163061514*x^6 - 444713220*x^4 + 466948881*x^2 - 54059887)
\( x^{18} - 126 x^{16} + 6615 x^{14} - 187278 x^{12} + 3090087 x^{10} - 29950074 x^{8} + 163061514 x^{6} + \cdots - 54059887 \)
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: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1225591470507623894379401358877860298752\)
\(\medspace = 2^{18}\cdot 3^{44}\cdot 7^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(148.45\) | 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: | $2\cdot 3^{22/9}7^{5/6}\approx 148.44814080346936$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{7}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(756=2^{2}\cdot 3^{3}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{756}(1,·)$, $\chi_{756}(115,·)$, $\chi_{756}(529,·)$, $\chi_{756}(277,·)$, $\chi_{756}(121,·)$, $\chi_{756}(25,·)$, $\chi_{756}(367,·)$, $\chi_{756}(607,·)$, $\chi_{756}(355,·)$, $\chi_{756}(103,·)$, $\chi_{756}(619,·)$, $\chi_{756}(559,·)$, $\chi_{756}(625,·)$, $\chi_{756}(307,·)$, $\chi_{756}(373,·)$, $\chi_{756}(55,·)$, $\chi_{756}(505,·)$, $\chi_{756}(253,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}$, $\frac{1}{7}a^{7}$, $\frac{1}{7}a^{8}$, $\frac{1}{2779}a^{9}-\frac{9}{397}a^{7}+\frac{189}{397}a^{5}+\frac{118}{397}a^{3}-\frac{89}{397}a$, $\frac{1}{897617}a^{10}+\frac{10259}{897617}a^{8}-\frac{63785}{897617}a^{6}+\frac{33069}{128231}a^{4}-\frac{10808}{128231}a^{2}+\frac{71}{323}$, $\frac{1}{897617}a^{11}-\frac{11}{128231}a^{9}-\frac{53772}{897617}a^{7}+\frac{3030}{128231}a^{5}+\frac{51854}{128231}a^{3}+\frac{50474}{128231}a$, $\frac{1}{6283319}a^{12}-\frac{4745}{897617}a^{8}-\frac{57450}{897617}a^{6}-\frac{31845}{128231}a^{4}+\frac{53191}{128231}a^{2}+\frac{135}{323}$, $\frac{1}{6283319}a^{13}+\frac{100}{897617}a^{9}+\frac{3144}{128231}a^{7}-\frac{13757}{128231}a^{5}-\frac{16254}{128231}a^{3}+\frac{7083}{128231}a$, $\frac{1}{6283319}a^{14}+\frac{21956}{897617}a^{8}-\frac{1118}{897617}a^{6}+\frac{10852}{128231}a^{4}+\frac{3265}{6749}a^{2}+\frac{6}{323}$, $\frac{1}{6283319}a^{15}-\frac{8}{897617}a^{9}-\frac{27927}{897617}a^{7}-\frac{36952}{128231}a^{5}+\frac{1837}{6749}a^{3}+\frac{33713}{128231}a$, $\frac{1}{6283319}a^{16}+\frac{455}{7543}a^{8}+\frac{26}{52801}a^{6}+\frac{2529}{7543}a^{4}-\frac{3103}{7543}a^{2}-\frac{78}{323}$, $\frac{1}{6283319}a^{17}-\frac{1}{7543}a^{9}-\frac{2539}{52801}a^{7}+\frac{2681}{7543}a^{5}-\frac{2609}{7543}a^{3}+\frac{53983}{128231}a$
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
$C_{3}$, which has order $3$ (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: | $17$ | 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: |
$\frac{39}{6283319}a^{12}-\frac{468}{897617}a^{10}+\frac{2106}{128231}a^{8}-\frac{215297}{897617}a^{6}+\frac{208245}{128231}a^{4}-\frac{561267}{128231}a^{2}+\frac{1107}{323}$, $\frac{39}{6283319}a^{12}-\frac{468}{897617}a^{10}+\frac{2106}{128231}a^{8}-\frac{215297}{897617}a^{6}+\frac{208245}{128231}a^{4}-\frac{561267}{128231}a^{2}+\frac{784}{323}$, $\frac{1}{330701}a^{14}-\frac{1846}{6283319}a^{12}+\frac{9900}{897617}a^{10}-\frac{179281}{897617}a^{8}+\frac{1589368}{897617}a^{6}-\frac{882462}{128231}a^{4}+\frac{1158479}{128231}a^{2}-\frac{334}{323}$, $\frac{2}{6283319}a^{16}-\frac{32}{897617}a^{14}+\frac{10247}{6283319}a^{12}-\frac{5001}{128231}a^{10}+\frac{27257}{52801}a^{8}-\frac{172931}{47243}a^{6}+\frac{1557233}{128231}a^{4}-\frac{88356}{6749}a^{2}+\frac{362}{323}$, $\frac{3}{6283319}a^{16}-\frac{317}{6283319}a^{14}+\frac{13527}{6283319}a^{12}-\frac{42802}{897617}a^{10}+\frac{526892}{897617}a^{8}-\frac{3578139}{897617}a^{6}+\frac{1773189}{128231}a^{4}-\frac{2555079}{128231}a^{2}+\frac{1808}{323}$, $\frac{1}{6283319}a^{14}-\frac{6}{330701}a^{12}+\frac{92}{128231}a^{10}-\frac{662}{52801}a^{8}+\frac{93012}{897617}a^{6}-\frac{67427}{128231}a^{4}+\frac{373933}{128231}a^{2}-\frac{2606}{323}$, $\frac{2}{6283319}a^{16}-\frac{206}{6283319}a^{14}+\frac{8327}{6283319}a^{12}-\frac{1249}{47243}a^{10}+\frac{239543}{897617}a^{8}-\frac{1092527}{897617}a^{6}+\frac{180306}{128231}a^{4}+\frac{357312}{128231}a^{2}-\frac{128}{323}$, $\frac{1}{6283319}a^{16}-\frac{16}{897617}a^{14}+\frac{303}{369607}a^{12}-\frac{17970}{897617}a^{10}+\frac{252267}{897617}a^{8}-\frac{2062530}{897617}a^{6}+\frac{1338567}{128231}a^{4}-\frac{2733828}{128231}a^{2}+\frac{488}{323}$, $\frac{3}{6283319}a^{15}-\frac{45}{897617}a^{13}+\frac{227}{6283319}a^{12}+\frac{270}{128231}a^{11}-\frac{2724}{897617}a^{10}-\frac{39779}{897617}a^{9}+\frac{12258}{128231}a^{8}+\frac{60336}{128231}a^{7}-\frac{1246563}{897617}a^{6}-\frac{266868}{128231}a^{5}+\frac{1172637}{128231}a^{4}+\frac{44168}{128231}a^{3}-\frac{2852577}{128231}a^{2}+\frac{1545159}{128231}a+\frac{2294}{323}$, $\frac{8}{6283319}a^{15}-\frac{120}{897617}a^{13}+\frac{30}{6283319}a^{12}+\frac{720}{128231}a^{11}-\frac{360}{897617}a^{10}-\frac{108446}{897617}a^{9}+\frac{1620}{128231}a^{8}+\frac{182214}{128231}a^{7}-\frac{175477}{897617}a^{6}-\frac{1159326}{128231}a^{5}+\frac{219372}{128231}a^{4}+\frac{3642465}{128231}a^{3}-\frac{1053171}{128231}a^{2}-\frac{4089267}{128231}a+\frac{3684}{323}$, $\frac{1}{6283319}a^{17}-\frac{1}{52801}a^{15}-\frac{20}{6283319}a^{14}+\frac{7}{7543}a^{13}+\frac{1944}{6283319}a^{12}-\frac{182}{7543}a^{11}-\frac{10588}{897617}a^{10}+\frac{18903}{52801}a^{9}+\frac{28536}{128231}a^{8}-\frac{22980}{7543}a^{7}-\frac{1931801}{897617}a^{6}+\frac{108024}{7543}a^{5}+\frac{1286683}{128231}a^{4}-\frac{257544}{7543}a^{3}-\frac{2549071}{128231}a^{2}+\frac{3968964}{128231}a+\frac{4116}{323}$, $\frac{1}{6283319}a^{17}-\frac{1}{52801}a^{15}+\frac{37}{6283319}a^{14}+\frac{7}{7543}a^{13}-\frac{3642}{6283319}a^{12}-\frac{182}{7543}a^{11}+\frac{20135}{897617}a^{10}+\frac{18827}{52801}a^{9}-\frac{55254}{128231}a^{8}-\frac{22296}{7543}a^{7}+\frac{3816193}{897617}a^{6}+\frac{93660}{7543}a^{5}-\frac{2548391}{128231}a^{4}-\frac{145824}{7543}a^{3}+\frac{4243106}{128231}a^{2}+\frac{108791}{128231}a-\frac{1470}{323}$, $\frac{55}{6283319}a^{12}-\frac{55}{897617}a^{11}-\frac{424}{897617}a^{10}+\frac{3589}{897617}a^{9}+\frac{325}{52801}a^{8}-\frac{79386}{897617}a^{7}+\frac{1796}{47243}a^{6}+\frac{95949}{128231}a^{5}-\frac{128549}{128231}a^{4}-\frac{235347}{128231}a^{3}+\frac{23968}{6749}a^{2}-\frac{25725}{128231}a-\frac{392}{323}$, $\frac{55}{6283319}a^{12}+\frac{16}{897617}a^{11}-\frac{871}{897617}a^{10}-\frac{1878}{897617}a^{9}+\frac{36068}{897617}a^{8}+\frac{77963}{897617}a^{7}-\frac{690649}{897617}a^{6}-\frac{201845}{128231}a^{5}+\frac{862021}{128231}a^{4}+\frac{1522822}{128231}a^{3}-\frac{2791985}{128231}a^{2}-\frac{3366545}{128231}a+\frac{196}{17}$, $\frac{5}{6283319}a^{16}-\frac{80}{897617}a^{14}-\frac{16}{6283319}a^{13}+\frac{1339}{330701}a^{12}+\frac{208}{897617}a^{11}-\frac{85772}{897617}a^{10}-\frac{6634}{897617}a^{9}+\frac{402}{323}a^{8}+\frac{11658}{128231}a^{7}-\frac{7852063}{897617}a^{6}-\frac{23363}{128231}a^{5}+\frac{3825435}{128231}a^{4}-\frac{353297}{128231}a^{3}-\frac{4816705}{128231}a^{2}+\frac{816389}{128231}a+\frac{2170}{323}$, $\frac{4}{6283319}a^{16}-\frac{64}{897617}a^{14}+\frac{39}{6283319}a^{13}+\frac{20423}{6283319}a^{12}-\frac{507}{897617}a^{11}-\frac{69460}{897617}a^{10}+\frac{17099}{897617}a^{9}+\frac{131466}{128231}a^{8}-\frac{36774}{128231}a^{7}-\frac{392305}{52801}a^{6}+\frac{224443}{128231}a^{5}+\frac{3435189}{128231}a^{4}-\frac{223412}{128231}a^{3}-\frac{4786706}{128231}a^{2}-\frac{1086820}{128231}a+\frac{434}{323}$, $\frac{3}{2779}a^{9}-\frac{27}{397}a^{7}+\frac{567}{397}a^{5}-\frac{4410}{397}a^{3}+\frac{9261}{397}a-8$
| 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: | \( 58951439744759.19 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 58951439744759.19 \cdot 3}{2\cdot\sqrt{1225591470507623894379401358877860298752}}\cr\approx \mathstrut & 0.662144433267692 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 126*x^16 + 6615*x^14 - 187278*x^12 + 3090087*x^10 - 29950074*x^8 + 163061514*x^6 - 444713220*x^4 + 466948881*x^2 - 54059887)
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 - 126*x^16 + 6615*x^14 - 187278*x^12 + 3090087*x^10 - 29950074*x^8 + 163061514*x^6 - 444713220*x^4 + 466948881*x^2 - 54059887, 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 - 126*x^16 + 6615*x^14 - 187278*x^12 + 3090087*x^10 - 29950074*x^8 + 163061514*x^6 - 444713220*x^4 + 466948881*x^2 - 54059887);
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 - 126*x^16 + 6615*x^14 - 187278*x^12 + 3090087*x^10 - 29950074*x^8 + 163061514*x^6 - 444713220*x^4 + 466948881*x^2 - 54059887);
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
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 cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{7}) \), \(\Q(\zeta_{9})^+\), 6.6.144027072.1, 9.9.3691950281939241.2 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | $18$ | R | $18$ | $18$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.1.0.1}{1} }^{18}$ | $18$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | $18$ | $18$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.18.115 | $x^{18} + 18 x^{17} + 162 x^{16} + 960 x^{15} + 4320 x^{14} + 16128 x^{13} + 53696 x^{12} + 165120 x^{11} + 449824 x^{10} + 1006400 x^{9} + 1826368 x^{8} + 2905088 x^{7} + 3317760 x^{6} - 418816 x^{5} - 6684672 x^{4} - 4984832 x^{3} + 2483456 x^{2} + 3566080 x + 1829376$ | $2$ | $9$ | $18$ | $C_{18}$ | $[2]^{9}$ |
|
\(3\)
| 3.9.22.6 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.6 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
|
\(7\)
| 7.18.15.2 | $x^{18} - 42 x^{12} - 1372$ | $6$ | $3$ | $15$ | $C_{18}$ | $[\ ]_{6}^{3}$ |