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Properties

Label 18.0.581...368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-5.814\times 10^{29}$
Root discriminant \(45.04\)
Ramified primes $2,3,43$
Class number $9$ (GRH)
Class group [3, 3] (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 24*x^16 + 180*x^14 + 101*x^12 - 6168*x^10 + 14616*x^8 + 39119*x^6 + 35820*x^4 - 19368*x^2 + 2107)
 
gp: K = bnfinit(y^18 - 24*y^16 + 180*y^14 + 101*y^12 - 6168*y^10 + 14616*y^8 + 39119*y^6 + 35820*y^4 - 19368*y^2 + 2107, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 24*x^16 + 180*x^14 + 101*x^12 - 6168*x^10 + 14616*x^8 + 39119*x^6 + 35820*x^4 - 19368*x^2 + 2107);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 24*x^16 + 180*x^14 + 101*x^12 - 6168*x^10 + 14616*x^8 + 39119*x^6 + 35820*x^4 - 19368*x^2 + 2107)
 

\( x^{18} - 24 x^{16} + 180 x^{14} + 101 x^{12} - 6168 x^{10} + 14616 x^{8} + 39119 x^{6} + 35820 x^{4} + \cdots + 2107 \) 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:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-581414905561488081454979002368\) \(\medspace = -\,2^{12}\cdot 3^{24}\cdot 43^{9}\) 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:  \(45.04\)
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:  $2^{2/3}3^{4/3}43^{1/2}\approx 45.03835964061842$
Ramified primes:   \(2\), \(3\), \(43\) 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:  \(\Q(\sqrt{-43}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{36}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{6}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{5}{36}$, $\frac{1}{36}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{6}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{5}{36}a$, $\frac{1}{85140}a^{14}+\frac{569}{85140}a^{12}-\frac{969}{4730}a^{10}+\frac{204}{2365}a^{8}-\frac{1}{2}a^{7}+\frac{4027}{14190}a^{6}-\frac{1}{2}a^{5}-\frac{6649}{14190}a^{4}-\frac{1}{2}a^{3}+\frac{6989}{85140}a^{2}-\frac{1}{2}a-\frac{959}{1980}$, $\frac{1}{595980}a^{15}+\frac{1916}{148995}a^{13}+\frac{5157}{66220}a^{11}-\frac{1}{4}a^{10}+\frac{7911}{66220}a^{9}-\frac{1}{4}a^{8}+\frac{15149}{198660}a^{7}+\frac{1}{4}a^{6}-\frac{62963}{198660}a^{5}-\frac{1}{4}a^{4}+\frac{14137}{297990}a^{3}+\frac{1}{4}a^{2}+\frac{2341}{13860}a+\frac{1}{4}$, $\frac{1}{3797409341880}a^{16}+\frac{4682131}{3797409341880}a^{14}-\frac{48542760599}{3797409341880}a^{12}-\frac{6798883345}{42193437132}a^{10}-\frac{24801037585}{126580311396}a^{8}+\frac{29008411463}{126580311396}a^{6}-\frac{1}{2}a^{5}+\frac{3624148859}{11904104520}a^{4}-\frac{1690162363369}{3797409341880}a^{2}-\frac{6222216019}{12615977880}$, $\frac{1}{3797409341880}a^{17}-\frac{337915}{759481868376}a^{15}+\frac{8108077447}{3797409341880}a^{13}+\frac{53905483}{4906213620}a^{11}-\frac{5912806847}{90414508140}a^{9}+\frac{255004959463}{632901556980}a^{7}-\frac{1}{2}a^{6}+\frac{90591355}{2380820904}a^{5}-\frac{1}{2}a^{4}+\frac{977741027597}{3797409341880}a^{3}-\frac{24128180539}{88311845160}a-\frac{1}{2}$ 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

$C_{3}\times C_{3}$, which has order $9$ (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:   \( -1 \)  (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:   $\frac{17387483}{992008710}a^{17}-\frac{107446417}{18706449960}a^{16}-\frac{165532855}{396803484}a^{15}+\frac{511417627}{3741289992}a^{14}+\frac{1523613478}{496004355}a^{13}-\frac{18825845629}{18706449960}a^{12}+\frac{131317982}{55111595}a^{11}-\frac{813273973}{1039247220}a^{10}-\frac{3235598839}{30060870}a^{9}+\frac{109963389143}{3117741660}a^{8}+\frac{38807816009}{165334785}a^{7}-\frac{239623605211}{3117741660}a^{6}+\frac{145325589149}{198401742}a^{5}-\frac{81683150567}{340117272}a^{4}+\frac{1535232837527}{1984017420}a^{3}-\frac{4752688740089}{18706449960}a^{2}-\frac{2140587811}{11534985}a+\frac{26561314693}{435033720}$, $\frac{54170033}{11904104520}a^{17}-\frac{257727917}{2380820904}a^{15}+\frac{220420967}{276839640}a^{13}+\frac{415608317}{661339140}a^{11}-\frac{5039296607}{180365220}a^{9}+\frac{120224224229}{1984017420}a^{7}+\frac{454826998541}{2380820904}a^{5}+\frac{2416879825411}{11904104520}a^{3}-\frac{13167581717}{276839640}a+\frac{1}{2}$, $\frac{4907479249}{3797409341880}a^{17}-\frac{343963}{1212455090}a^{16}-\frac{116645547287}{3797409341880}a^{15}+\frac{29495227}{4364838324}a^{14}+\frac{171252824051}{759481868376}a^{13}-\frac{271190932}{5456047905}a^{12}+\frac{38725280089}{210967185660}a^{11}-\frac{24739733}{606227545}a^{10}-\frac{5023818535049}{632901556980}a^{9}+\frac{1068633371}{606227545}a^{8}+\frac{1543253142469}{90414508140}a^{7}-\frac{13798512877}{3637365270}a^{6}+\frac{652182700847}{11904104520}a^{5}-\frac{820113289}{66133914}a^{4}+\frac{43787296873645}{759481868376}a^{3}-\frac{252760314743}{21824191620}a^{2}-\frac{1211697663343}{88311845160}a+\frac{49329952}{18126405}$, $\frac{4907479249}{3797409341880}a^{17}+\frac{343963}{1212455090}a^{16}-\frac{116645547287}{3797409341880}a^{15}-\frac{29495227}{4364838324}a^{14}+\frac{171252824051}{759481868376}a^{13}+\frac{271190932}{5456047905}a^{12}+\frac{38725280089}{210967185660}a^{11}+\frac{24739733}{606227545}a^{10}-\frac{5023818535049}{632901556980}a^{9}-\frac{1068633371}{606227545}a^{8}+\frac{1543253142469}{90414508140}a^{7}+\frac{13798512877}{3637365270}a^{6}+\frac{652182700847}{11904104520}a^{5}+\frac{820113289}{66133914}a^{4}+\frac{43787296873645}{759481868376}a^{3}+\frac{252760314743}{21824191620}a^{2}-\frac{1211697663343}{88311845160}a-\frac{49329952}{18126405}$, $\frac{14193907}{2739833580}a^{17}+\frac{1538265083}{759481868376}a^{16}-\frac{1773498079}{14384126295}a^{15}-\frac{183092375729}{3797409341880}a^{14}+\frac{474854689}{523059138}a^{13}+\frac{1349159277289}{3797409341880}a^{12}+\frac{4502180259}{6392945020}a^{11}+\frac{14326523404}{52741796415}a^{10}-\frac{203344866233}{6392945020}a^{9}-\frac{1966354188494}{158225389245}a^{8}+\frac{120933456893}{1743530460}a^{7}+\frac{4305629105548}{158225389245}a^{6}+\frac{11924749611}{55111595}a^{5}+\frac{1001150157239}{11904104520}a^{4}+\frac{239688125521}{1046118276}a^{3}+\frac{339917098803749}{3797409341880}a^{2}-\frac{71800051919}{1338058260}a-\frac{264455815117}{12615977880}$, $\frac{3715383173}{632901556980}a^{17}-\frac{6847930171}{3797409341880}a^{16}-\frac{9820430537}{70322395220}a^{15}+\frac{163090765031}{3797409341880}a^{14}+\frac{649994954159}{632901556980}a^{13}-\frac{1202239282273}{3797409341880}a^{12}+\frac{57216097693}{70322395220}a^{11}-\frac{12803359252}{52741796415}a^{10}-\frac{1086953322659}{30138169380}a^{9}+\frac{1755823487672}{158225389245}a^{8}+\frac{5497824732577}{70322395220}a^{7}-\frac{7701228654083}{316450778490}a^{6}+\frac{122568327652}{496004355}a^{5}-\frac{898407892931}{11904104520}a^{4}+\frac{9130614049033}{35161197610}a^{3}-\frac{289862862961043}{3797409341880}a^{2}-\frac{93266711465}{1471864086}a+\frac{47417083205}{2523195576}$, $\frac{554099320199}{253160622792}a^{17}-\frac{31185881347}{31645077849}a^{16}-\frac{4395404269909}{84386874264}a^{15}+\frac{742338320456}{31645077849}a^{14}+\frac{97069010974343}{253160622792}a^{13}-\frac{5467807762600}{31645077849}a^{12}+\frac{600018642889}{2009211292}a^{11}-\frac{468932553248}{3516119761}a^{10}-\frac{567047757101671}{42193437132}a^{9}+\frac{63831684461344}{10548359283}a^{8}+\frac{411657950475217}{14064479044}a^{7}-\frac{139385522832104}{10548359283}a^{6}+\frac{72646784324497}{793606968}a^{5}-\frac{4078590996476}{99200871}a^{4}+\frac{11\!\cdots\!45}{12055267752}a^{3}-\frac{13\!\cdots\!76}{31645077849}a^{2}-\frac{134845903726607}{5887456344}a+\frac{2161863949589}{210266298}$, $\frac{158357053}{28128958088}a^{17}+\frac{26886583}{10548359283}a^{16}-\frac{33926850973}{253160622792}a^{15}-\frac{213811116}{3516119761}a^{14}+\frac{27737610613}{28128958088}a^{13}+\frac{4741588616}{10548359283}a^{12}+\frac{1573804267}{2009211292}a^{11}+\frac{1117122792}{3516119761}a^{10}-\frac{488095829975}{14064479044}a^{9}-\frac{54537094798}{3516119761}a^{8}+\frac{3189584250089}{42193437132}a^{7}+\frac{119345738112}{3516119761}a^{6}+\frac{20820642907}{88178552}a^{5}+\frac{3489410252}{33066957}a^{4}+\frac{9021006813769}{36165803256}a^{3}+\frac{392122226772}{3516119761}a^{2}-\frac{39294602685}{654161816}a-\frac{1882437943}{70088766}$ 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:  \( 11837492.6514 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 11837492.6514 \cdot 9}{2\cdot\sqrt{581414905561488081454979002368}}\cr\approx \mathstrut & 1.06622281345 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 24*x^16 + 180*x^14 + 101*x^12 - 6168*x^10 + 14616*x^8 + 39119*x^6 + 35820*x^4 - 19368*x^2 + 2107)
 
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 - 24*x^16 + 180*x^14 + 101*x^12 - 6168*x^10 + 14616*x^8 + 39119*x^6 + 35820*x^4 - 19368*x^2 + 2107, 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 - 24*x^16 + 180*x^14 + 101*x^12 - 6168*x^10 + 14616*x^8 + 39119*x^6 + 35820*x^4 - 19368*x^2 + 2107);
 
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 - 24*x^16 + 180*x^14 + 101*x^12 - 6168*x^10 + 14616*x^8 + 39119*x^6 + 35820*x^4 - 19368*x^2 + 2107);
 
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

$C_3:S_3$ (as 18T4):

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 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-43}) \), 3.1.13932.1 x3, 3.1.13932.2 x3, 3.1.3483.1 x3, 3.1.172.1 x3, 6.0.8346326832.1, 6.0.8346326832.2, 6.0.521645427.1, 6.0.1272112.1, 9.1.116281025423424.4 x9

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 sibling: 9.1.116281025423424.4
Minimal sibling: 9.1.116281025423424.4

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 ${\href{/padicField/5.2.0.1}{2} }^{9}$ ${\href{/padicField/7.2.0.1}{2} }^{9}$ ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.3.0.1}{3} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{9}$ ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{9}$ ${\href{/padicField/41.3.0.1}{3} }^{6}$ R ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.3.0.1}{3} }^{6}$ ${\href{/padicField/59.1.0.1}{1} }^{18}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(3\) Copy content Toggle raw display 3.6.8.5$x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$$3$$2$$8$$S_3$$[2]^{2}$
\(43\) Copy content Toggle raw display 43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$