Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 22*x^15 + 250*x^12 - 1634*x^9 + 9425*x^6 - 27040*x^3 + 54080)
gp: K = bnfinit(y^18 - 22*y^15 + 250*y^12 - 1634*y^9 + 9425*y^6 - 27040*y^3 + 54080, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 22*x^15 + 250*x^12 - 1634*x^9 + 9425*x^6 - 27040*x^3 + 54080);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 22*x^15 + 250*x^12 - 1634*x^9 + 9425*x^6 - 27040*x^3 + 54080)
\( x^{18} - 22x^{15} + 250x^{12} - 1634x^{9} + 9425x^{6} - 27040x^{3} + 54080 \)
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: |
\(-291253980508086591590400000000\)
\(\medspace = -\,2^{18}\cdot 3^{20}\cdot 5^{8}\cdot 13^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(43.34\) | 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: | $2\cdot 3^{3/2}5^{8/9}13^{8/9}\approx 424.80577976779654$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{52}a^{14}+\frac{1}{13}a^{11}-\frac{5}{26}a^{8}+\frac{1}{13}a^{5}+\frac{1}{4}a^{2}$, $\frac{1}{17869805720}a^{15}-\frac{1553064407}{8934902860}a^{12}+\frac{2187565999}{8934902860}a^{9}+\frac{606947903}{1786980572}a^{6}+\frac{59692365}{274920088}a^{3}-\frac{16266472}{34365011}$, $\frac{1}{107218834320}a^{16}-\frac{1}{53609417160}a^{15}-\frac{1}{156}a^{14}-\frac{1553064407}{53609417160}a^{13}+\frac{6020515837}{26804708580}a^{12}-\frac{5}{26}a^{11}-\frac{11214788291}{53609417160}a^{10}-\frac{2187565999}{26804708580}a^{9}+\frac{5}{78}a^{8}+\frac{2393928475}{10721883432}a^{7}+\frac{2073522955}{5360941716}a^{6}-\frac{5}{26}a^{5}-\frac{627607855}{1649520528}a^{4}+\frac{215227723}{824760264}a^{3}-\frac{1}{12}a^{2}+\frac{86828561}{206190066}a-\frac{18098539}{103095033}$, $\frac{1}{214437668640}a^{17}+\frac{1}{53609417160}a^{15}-\frac{59487989}{35739611440}a^{14}-\frac{1}{6}a^{13}-\frac{1553064407}{26804708580}a^{12}-\frac{23586192251}{107218834320}a^{11}+\frac{2187565999}{26804708580}a^{9}-\frac{118424135}{7147922288}a^{8}+\frac{1}{6}a^{7}+\frac{2393928475}{5360941716}a^{6}-\frac{13107463699}{42887533728}a^{5}-\frac{215227723}{824760264}a^{3}+\frac{51852861}{137460044}a^{2}+\frac{1}{3}a-\frac{16266472}{103095033}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{971}{6705368} a^{15} - \frac{10391}{3352684} a^{12} + \frac{109989}{3352684} a^{9} - \frac{570505}{3352684} a^{6} + \frac{5552495}{6705368} a^{3} - \frac{1162142}{838171} \)
(order $4$)
| 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: |
$\frac{1553307}{8934902860}a^{15}-\frac{7990552}{2233725715}a^{12}+\frac{151744973}{4467451430}a^{9}-\frac{62022984}{446745143}a^{6}+\frac{57301867}{137460044}a^{3}-\frac{4065341}{34365011}$, $\frac{732111}{17869805720}a^{15}-\frac{5172447}{8934902860}a^{12}+\frac{501759}{8934902860}a^{9}-\frac{33291145}{1786980572}a^{6}+\frac{2383991}{274920088}a^{3}-\frac{12170452}{34365011}$, $\frac{706765}{7147922288}a^{17}-\frac{2004707}{8247602640}a^{16}+\frac{1391449}{10721883432}a^{15}-\frac{27938275}{10721883432}a^{14}+\frac{20073649}{4123801320}a^{13}-\frac{14491567}{5360941716}a^{12}+\frac{110959033}{3573961144}a^{11}-\frac{193062203}{4123801320}a^{10}+\frac{105285769}{5360941716}a^{9}-\frac{2052936011}{10721883432}a^{8}+\frac{162334675}{824760264}a^{7}+\frac{136315219}{5360941716}a^{6}+\frac{5293465981}{7147922288}a^{5}-\frac{1075213039}{1649520528}a^{4}-\frac{564298595}{824760264}a^{3}-\frac{629178419}{412380132}a^{2}-\frac{152251891}{206190066}a+\frac{232840307}{103095033}$, $\frac{4266767}{107218834320}a^{17}+\frac{796585}{3573961144}a^{16}-\frac{20167463}{53609417160}a^{15}-\frac{42515123}{17869805720}a^{14}-\frac{44153179}{5360941716}a^{13}-\frac{22712009}{26804708580}a^{12}+\frac{857210303}{53609417160}a^{11}+\frac{115115041}{1786980572}a^{10}-\frac{930737177}{26804708580}a^{9}-\frac{602462517}{3573961144}a^{8}-\frac{2491187405}{5360941716}a^{7}+\frac{423380413}{5360941716}a^{6}+\frac{12503932843}{21443766864}a^{5}+\frac{557905621}{274920088}a^{4}-\frac{662818291}{824760264}a^{3}-\frac{83551675}{34365011}a^{2}-\frac{447937252}{103095033}a+\frac{334823018}{103095033}$, $\frac{23479791}{35739611440}a^{17}+\frac{94876553}{107218834320}a^{16}-\frac{1803659}{1030950330}a^{15}-\frac{869868901}{53609417160}a^{14}-\frac{971436391}{53609417160}a^{13}+\frac{17121113}{515475165}a^{12}+\frac{3336150079}{17869805720}a^{11}+\frac{10336078637}{53609417160}a^{10}-\frac{309022987}{1030950330}a^{9}-\frac{12265205317}{10721883432}a^{8}-\frac{11323881541}{10721883432}a^{7}+\frac{137727449}{103095033}a^{6}+\frac{2813845687}{549840176}a^{5}+\frac{7894143721}{1649520528}a^{4}-\frac{850078139}{103095033}a^{3}-\frac{5605025695}{412380132}a^{2}-\frac{25674299}{206190066}a+\frac{1967489452}{103095033}$, $\frac{1193691}{5498401760}a^{17}-\frac{4645977}{17869805720}a^{16}-\frac{2005261}{4467451430}a^{15}-\frac{172552761}{35739611440}a^{14}+\frac{88659029}{8934902860}a^{13}+\frac{7584429}{4467451430}a^{12}+\frac{1522137107}{35739611440}a^{11}-\frac{793308903}{8934902860}a^{10}+\frac{70449697}{4467451430}a^{9}-\frac{2256131467}{7147922288}a^{8}+\frac{960453487}{1786980572}a^{7}+\frac{109103019}{893490286}a^{6}+\frac{14683332707}{14295844576}a^{5}-\frac{691974637}{274920088}a^{4}+\frac{47273753}{34365011}a^{3}-\frac{661542401}{137460044}a^{2}+\frac{253222626}{34365011}a-\frac{65102097}{34365011}$, $\frac{46929049}{53609417160}a^{17}+\frac{29049713}{26804708580}a^{16}+\frac{6007207}{6701177145}a^{15}-\frac{128256842}{6701177145}a^{14}-\frac{164124853}{6701177145}a^{13}-\frac{105372437}{4467451430}a^{12}+\frac{5555032291}{26804708580}a^{11}+\frac{3615735857}{13402354290}a^{10}+\frac{3796118977}{13402354290}a^{9}-\frac{6677223419}{5360941716}a^{8}-\frac{2246593291}{1340235429}a^{7}-\frac{1768514847}{893490286}a^{6}+\frac{69118303649}{10721883432}a^{5}+\frac{3561846349}{412380132}a^{4}+\frac{1969054021}{206190066}a^{3}-\frac{6300939829}{412380132}a^{2}-\frac{2443174688}{103095033}a-\frac{1162084970}{34365011}$, $\frac{6742121}{214437668640}a^{17}-\frac{9768493}{53609417160}a^{16}+\frac{296801}{6701177145}a^{15}-\frac{93009907}{107218834320}a^{14}+\frac{63065911}{26804708580}a^{13}+\frac{14738619}{4467451430}a^{12}+\frac{475595069}{107218834320}a^{11}-\frac{178830487}{26804708580}a^{10}-\frac{1006507969}{13402354290}a^{9}+\frac{1209844835}{21443766864}a^{8}-\frac{685359683}{5360941716}a^{7}+\frac{629329693}{893490286}a^{6}-\frac{30387441323}{42887533728}a^{5}+\frac{507151879}{824760264}a^{4}-\frac{427418425}{206190066}a^{3}+\frac{265153913}{206190066}a^{2}-\frac{356815852}{103095033}a+\frac{185611544}{34365011}$
| 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: | \( 139848639.045 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 139848639.045 \cdot 3}{4\cdot\sqrt{291253980508086591590400000000}}\cr\approx \mathstrut & 2.96621337058 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 22*x^15 + 250*x^12 - 1634*x^9 + 9425*x^6 - 27040*x^3 + 54080)
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 - 22*x^15 + 250*x^12 - 1634*x^9 + 9425*x^6 - 27040*x^3 + 54080, 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 - 22*x^15 + 250*x^12 - 1634*x^9 + 9425*x^6 - 27040*x^3 + 54080);
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 - 22*x^15 + 250*x^12 - 1634*x^9 + 9425*x^6 - 27040*x^3 + 54080);
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
$C_3^4.S_3^2$ (as 18T413):
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 2916 |
| The 58 conjugacy class representatives for $C_3^4.S_3^2$ are not computed |
| Character table for $C_3^4.S_3^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 6.0.2433600.1 |
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 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 | R | R | R | $18$ | $18$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $18$ | $18$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $18$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
|
\(3\)
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.12.18.75 | $x^{12} + 12 x^{10} + 36 x^{8} - 30 x^{6} - 180 x^{4} + 549$ | $6$ | $2$ | $18$ | $C_6\times C_2$ | $[2]_{2}^{2}$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.9.8.1 | $x^{9} + 5$ | $9$ | $1$ | $8$ | $(C_9:C_3):C_2$ | $[\ ]_{9}^{6}$ | |
|
\(13\)
| 13.9.8.3 | $x^{9} + 52$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |