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Properties

Label 18.18.318...336.1
Degree $18$
Signature $[18, 0]$
Discriminant $3.182\times 10^{35}$
Root discriminant \(93.84\)
Ramified primes $2,3,73$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^2:A_4$ (as 18T48)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 63*x^16 + 1221*x^14 - 9772*x^12 + 38544*x^10 - 84315*x^8 + 107675*x^6 - 79935*x^4 + 31974*x^2 - 5329)
 
gp: K = bnfinit(y^18 - 63*y^16 + 1221*y^14 - 9772*y^12 + 38544*y^10 - 84315*y^8 + 107675*y^6 - 79935*y^4 + 31974*y^2 - 5329, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 63*x^16 + 1221*x^14 - 9772*x^12 + 38544*x^10 - 84315*x^8 + 107675*x^6 - 79935*x^4 + 31974*x^2 - 5329);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 63*x^16 + 1221*x^14 - 9772*x^12 + 38544*x^10 - 84315*x^8 + 107675*x^6 - 79935*x^4 + 31974*x^2 - 5329)
 

\( x^{18} - 63 x^{16} + 1221 x^{14} - 9772 x^{12} + 38544 x^{10} - 84315 x^{8} + 107675 x^{6} - 79935 x^{4} + \cdots - 5329 \) Copy content Toggle raw display

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:   \(318184219227442073683553705831694336\) \(\medspace = 2^{18}\cdot 3^{24}\cdot 73^{10}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(93.84\)
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:  $2^{3/2}3^{4/3}73^{5/6}\approx 436.99424398700404$
Ramified primes:   \(2\), \(3\), \(73\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{292}a^{12}+\frac{5}{146}a^{10}+\frac{53}{292}a^{8}+\frac{5}{146}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}$, $\frac{1}{292}a^{13}+\frac{5}{146}a^{11}+\frac{53}{292}a^{9}+\frac{5}{146}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a$, $\frac{1}{292}a^{14}-\frac{47}{292}a^{10}+\frac{16}{73}a^{8}+\frac{23}{146}a^{6}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{292}a^{15}-\frac{47}{292}a^{11}+\frac{16}{73}a^{9}+\frac{23}{146}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{52268}a^{16}-\frac{15}{13067}a^{14}-\frac{33}{52268}a^{12}-\frac{2602}{13067}a^{10}+\frac{40}{13067}a^{8}+\frac{953}{26134}a^{6}+\frac{87}{716}a^{4}-\frac{59}{358}a^{2}+\frac{21}{179}$, $\frac{1}{52268}a^{17}-\frac{15}{13067}a^{15}-\frac{33}{52268}a^{13}-\frac{2602}{13067}a^{11}+\frac{40}{13067}a^{9}+\frac{953}{26134}a^{7}+\frac{87}{716}a^{5}-\frac{59}{358}a^{3}+\frac{21}{179}a$ Copy content Toggle raw display

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

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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{5}{73}a^{16}-\frac{313}{73}a^{14}+\frac{11959}{146}a^{12}-\frac{92899}{146}a^{10}+\frac{347573}{146}a^{8}-\frac{349373}{73}a^{6}+5329a^{4}-\frac{6205}{2}a^{2}+735$, $\frac{5}{73}a^{16}-\frac{313}{73}a^{14}+\frac{11959}{146}a^{12}-\frac{92899}{146}a^{10}+\frac{347573}{146}a^{8}-\frac{349373}{73}a^{6}+5329a^{4}-\frac{6205}{2}a^{2}+736$, $\frac{427}{13067}a^{16}-\frac{26694}{13067}a^{14}+\frac{508410}{13067}a^{12}-\frac{3925116}{13067}a^{10}+\frac{14533668}{13067}a^{8}-\frac{28768752}{13067}a^{6}+\frac{428443}{179}a^{4}-\frac{240663}{179}a^{2}+\frac{54305}{179}$, $\frac{427}{13067}a^{16}-\frac{26694}{13067}a^{14}+\frac{508410}{13067}a^{12}-\frac{3925116}{13067}a^{10}+\frac{14533668}{13067}a^{8}-\frac{28768752}{13067}a^{6}+\frac{428443}{179}a^{4}-\frac{240663}{179}a^{2}+\frac{54484}{179}$, $\frac{427}{13067}a^{16}-\frac{26694}{13067}a^{14}+\frac{508410}{13067}a^{12}-\frac{3925116}{13067}a^{10}+\frac{14533668}{13067}a^{8}-\frac{28768752}{13067}a^{6}+\frac{428443}{179}a^{4}-\frac{240663}{179}a^{2}+a+\frac{54484}{179}$, $\frac{427}{13067}a^{16}-\frac{26694}{13067}a^{14}+\frac{508410}{13067}a^{12}-\frac{3925116}{13067}a^{10}+\frac{14533668}{13067}a^{8}-\frac{28768752}{13067}a^{6}+\frac{428443}{179}a^{4}-\frac{240663}{179}a^{2}+a+\frac{54305}{179}$, $\frac{32681}{13067}a^{16}-\frac{2034787}{13067}a^{14}+\frac{76801565}{26134}a^{12}-\frac{290948562}{13067}a^{10}+\frac{2087288415}{26134}a^{8}-\frac{3952919919}{26134}a^{6}+\frac{55733641}{358}a^{4}-\frac{14717183}{179}a^{2}+\frac{3118777}{179}$, $\frac{29239}{26134}a^{16}-\frac{1823255}{26134}a^{14}+\frac{17263348}{13067}a^{12}-\frac{131709185}{13067}a^{10}+\frac{477873384}{13067}a^{8}-\frac{919018204}{13067}a^{6}+\frac{13209317}{179}a^{4}-\frac{14282387}{358}a^{2}+\frac{3111037}{358}$, $\frac{73177}{52268}a^{16}-\frac{1139899}{13067}a^{14}+\frac{43096959}{26134}a^{12}-\frac{327611623}{26134}a^{10}+\frac{2362215853}{52268}a^{8}-\frac{2249718543}{26134}a^{6}+\frac{63792105}{716}a^{4}-\frac{8463191}{179}a^{2}+\frac{7200999}{716}$, $\frac{36455}{52268}a^{16}-\frac{568663}{13067}a^{14}+\frac{10784045}{13067}a^{12}-\frac{82519409}{13067}a^{10}+\frac{1203747161}{52268}a^{8}-\frac{1166202679}{26134}a^{6}+\frac{33863283}{716}a^{4}-\frac{9268959}{358}a^{2}+\frac{4096661}{716}$, $\frac{76593}{52268}a^{17}-\frac{47509}{52268}a^{16}-\frac{4759723}{52268}a^{15}+\frac{2954897}{52268}a^{14}+\frac{22360434}{13067}a^{13}-\frac{27816811}{26134}a^{12}-\frac{671850767}{52268}a^{11}+\frac{419510441}{52268}a^{10}+\frac{2377006875}{52268}a^{9}-\frac{1493252931}{52268}a^{8}-\frac{2212313833}{26134}a^{7}+\frac{1400037943}{26134}a^{6}+\frac{61178399}{716}a^{5}-\frac{39025037}{716}a^{4}-\frac{31640125}{716}a^{3}+\frac{20347965}{716}a^{2}+\frac{6560007}{716}a-\frac{4252991}{716}$, $\frac{5768}{13067}a^{17}+\frac{447677}{52268}a^{16}-\frac{1425669}{52268}a^{15}-\frac{27808067}{52268}a^{14}+\frac{6613983}{13067}a^{13}+\frac{522050881}{52268}a^{12}-\frac{193931507}{52268}a^{11}-\frac{3913973315}{52268}a^{10}+\frac{165682714}{13067}a^{9}+\frac{3451753938}{13067}a^{8}-\frac{299394877}{13067}a^{7}-\frac{6403598147}{13067}a^{6}+\frac{8194599}{358}a^{5}+\frac{352985857}{716}a^{4}-\frac{8628959}{716}a^{3}-\frac{182011439}{716}a^{2}+\frac{935551}{358}a+\frac{18823377}{358}$, $\frac{157709}{52268}a^{17}-\frac{8670}{13067}a^{16}-\frac{4877871}{26134}a^{15}+\frac{549019}{13067}a^{14}+\frac{90720973}{26134}a^{13}-\frac{10746197}{13067}a^{12}-\frac{667346677}{26134}a^{11}+\frac{174253933}{26134}a^{10}+\frac{4566029723}{52268}a^{9}-\frac{686403145}{26134}a^{8}-\frac{4077375801}{26134}a^{7}+\frac{713666458}{13067}a^{6}+\frac{107785183}{716}a^{5}-\frac{21800557}{358}a^{4}-\frac{26622365}{358}a^{3}+\frac{6131183}{179}a^{2}+\frac{10555575}{716}a-\frac{1362298}{179}$, $\frac{86035}{26134}a^{17}-\frac{96811}{26134}a^{16}-\frac{10710661}{52268}a^{15}+\frac{12047815}{52268}a^{14}+\frac{50505906}{13067}a^{13}-\frac{227052429}{52268}a^{12}-\frac{1529282993}{52268}a^{11}+\frac{1715533649}{52268}a^{10}+\frac{2740862641}{26134}a^{9}-\frac{6127055251}{52268}a^{8}-\frac{5195558141}{26134}a^{7}+\frac{2886026858}{13067}a^{6}+\frac{73528527}{358}a^{5}-\frac{40491983}{179}a^{4}-\frac{78247813}{716}a^{3}+\frac{85231939}{716}a^{2}+\frac{4193967}{179}a-\frac{18036885}{716}$, $\frac{159983}{13067}a^{17}-\frac{159666}{13067}a^{16}-\frac{19894091}{26134}a^{15}+\frac{9927220}{13067}a^{14}+\frac{187139654}{13067}a^{13}-\frac{186761685}{13067}a^{12}-\frac{1409104961}{13067}a^{11}+\frac{19262741}{179}a^{10}+\frac{10010319053}{26134}a^{9}-\frac{4994275816}{13067}a^{8}-\frac{18731030901}{26134}a^{7}+\frac{18687561263}{26134}a^{6}+\frac{130318462}{179}a^{5}-\frac{259981579}{358}a^{4}-\frac{135797689}{358}a^{3}+\frac{67712779}{179}a^{2}+\frac{28392737}{358}a-\frac{28308693}{358}$, $\frac{125347}{26134}a^{17}+\frac{230417}{52268}a^{16}-\frac{3898956}{13067}a^{15}-\frac{7159351}{26134}a^{14}+\frac{146890861}{26134}a^{13}+\frac{67266942}{13067}a^{12}-\frac{1108804729}{26134}a^{11}-\frac{1010646207}{26134}a^{10}+\frac{3953764711}{26134}a^{9}+\frac{7154579223}{52268}a^{8}-\frac{7432061481}{26134}a^{7}-\frac{3334108572}{13067}a^{6}+\frac{103912515}{358}a^{5}+\frac{184916019}{716}a^{4}-\frac{27198243}{179}a^{3}-\frac{24016117}{179}a^{2}+\frac{11423969}{358}a+\frac{20041493}{716}$, $\frac{622863}{52268}a^{17}-\frac{235281}{52268}a^{16}-\frac{19335839}{26134}a^{15}+\frac{3680112}{13067}a^{14}+\frac{362620841}{26134}a^{13}-\frac{140399829}{26134}a^{12}-\frac{1356633738}{13067}a^{11}+\frac{543053462}{13067}a^{10}+\frac{19090278051}{52268}a^{9}-\frac{8036798571}{52268}a^{8}-\frac{17668760903}{26134}a^{7}+\frac{3938288521}{13067}a^{6}+\frac{486730767}{716}a^{5}-\frac{229692165}{716}a^{4}-\frac{125739483}{358}a^{3}+\frac{31280225}{179}a^{2}+\frac{52268045}{716}a-\frac{27269253}{716}$ Copy content Toggle raw display (assuming GRH)
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:  \( 1494891798290 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 1494891798290 \cdot 1}{2\cdot\sqrt{318184219227442073683553705831694336}}\cr\approx \mathstrut & 0.347360448788938 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 63*x^16 + 1221*x^14 - 9772*x^12 + 38544*x^10 - 84315*x^8 + 107675*x^6 - 79935*x^4 + 31974*x^2 - 5329)
 
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 - 63*x^16 + 1221*x^14 - 9772*x^12 + 38544*x^10 - 84315*x^8 + 107675*x^6 - 79935*x^4 + 31974*x^2 - 5329, 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 - 63*x^16 + 1221*x^14 - 9772*x^12 + 38544*x^10 - 84315*x^8 + 107675*x^6 - 79935*x^4 + 31974*x^2 - 5329);
 
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 - 63*x^16 + 1221*x^14 - 9772*x^12 + 38544*x^10 - 84315*x^8 + 107675*x^6 - 79935*x^4 + 31974*x^2 - 5329);
 
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

$C_3^2:A_4$ (as 18T48):

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 108
The 20 conjugacy class representatives for $C_3^2:A_4$
Character table for $C_3^2:A_4$

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.6.2237668416.1, 9.9.15091989595281.1

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 ${\href{/padicField/5.3.0.1}{3} }^{6}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{6}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.6.6.1$x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$$2$$3$$6$$A_4$$[2, 2]^{3}$
\(3\) Copy content Toggle raw display 3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
\(73\) Copy content Toggle raw display 73.6.5.3$x^{6} + 292$$6$$1$$5$$C_6$$[\ ]_{6}$
73.6.5.6$x^{6} + 949$$6$$1$$5$$C_6$$[\ ]_{6}$
73.6.0.1$x^{6} + 45 x^{3} + 23 x^{2} + 48 x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$