Properties

Label 18.0.435...963.1
Degree $18$
Signature $[0, 9]$
Discriminant $-4.356\times 10^{19}$
Root discriminant \(12.33\)
Ramified primes $3,19$
Class number $1$
Class group trivial
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 4*x^16 - x^15 + 12*x^14 - 9*x^13 + 12*x^12 - 18*x^11 + 26*x^10 - 12*x^9 - 4*x^8 - 5*x^7 + 31*x^6 - 17*x^5 - 3*x^4 + 5*x^2 - 3*x + 1)
 
gp: K = bnfinit(y^18 - y^17 + 4*y^16 - y^15 + 12*y^14 - 9*y^13 + 12*y^12 - 18*y^11 + 26*y^10 - 12*y^9 - 4*y^8 - 5*y^7 + 31*y^6 - 17*y^5 - 3*y^4 + 5*y^2 - 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 + 4*x^16 - x^15 + 12*x^14 - 9*x^13 + 12*x^12 - 18*x^11 + 26*x^10 - 12*x^9 - 4*x^8 - 5*x^7 + 31*x^6 - 17*x^5 - 3*x^4 + 5*x^2 - 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 + 4*x^16 - x^15 + 12*x^14 - 9*x^13 + 12*x^12 - 18*x^11 + 26*x^10 - 12*x^9 - 4*x^8 - 5*x^7 + 31*x^6 - 17*x^5 - 3*x^4 + 5*x^2 - 3*x + 1)
 

\( x^{18} - x^{17} + 4 x^{16} - x^{15} + 12 x^{14} - 9 x^{13} + 12 x^{12} - 18 x^{11} + 26 x^{10} - 12 x^{9} - 4 x^{8} - 5 x^{7} + 31 x^{6} - 17 x^{5} - 3 x^{4} + 5 x^{2} - 3 x + 1 \) 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:   \(-43564677551979246963\) \(\medspace = -\,3^{9}\cdot 19^{12}\) 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:  \(12.33\)
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:  $3^{1/2}19^{2/3}\approx 12.332838034173273$
Ramified primes:   \(3\), \(19\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:  $18$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{78}a^{15}-\frac{3}{26}a^{14}+\frac{3}{26}a^{13}-\frac{1}{78}a^{12}+\frac{6}{13}a^{11}-\frac{17}{78}a^{10}+\frac{35}{78}a^{9}+\frac{1}{26}a^{8}+\frac{1}{39}a^{7}+\frac{5}{39}a^{6}+\frac{35}{78}a^{5}+\frac{11}{78}a^{4}-\frac{8}{39}a^{3}-\frac{23}{78}a^{2}+\frac{7}{39}a-\frac{7}{78}$, $\frac{1}{858}a^{16}-\frac{2}{429}a^{15}-\frac{19}{143}a^{14}-\frac{4}{429}a^{13}+\frac{83}{858}a^{12}+\frac{397}{858}a^{11}+\frac{9}{143}a^{10}-\frac{70}{143}a^{9}-\frac{133}{286}a^{8}-\frac{185}{429}a^{7}-\frac{23}{78}a^{6}-\frac{37}{429}a^{5}-\frac{1}{2}a^{4}-\frac{11}{26}a^{3}+\frac{107}{858}a^{2}-\frac{145}{858}a-\frac{347}{858}$, $\frac{1}{1570998}a^{17}-\frac{7}{40282}a^{16}-\frac{8135}{1570998}a^{15}-\frac{156595}{1570998}a^{14}-\frac{2286}{20141}a^{13}+\frac{256579}{1570998}a^{12}-\frac{616127}{1570998}a^{11}+\frac{11575}{1570998}a^{10}+\frac{289771}{785499}a^{9}+\frac{177115}{785499}a^{8}+\frac{56867}{523666}a^{7}-\frac{122825}{523666}a^{6}-\frac{29609}{261833}a^{5}-\frac{69749}{142818}a^{4}-\frac{177217}{785499}a^{3}-\frac{170047}{1570998}a^{2}-\frac{254560}{785499}a+\frac{9201}{261833}$ Copy content Toggle raw display

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 $1$

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:   \( \frac{16945}{142818} a^{17} - \frac{23327}{71409} a^{16} + \frac{36250}{71409} a^{15} - \frac{21461}{23803} a^{14} + \frac{47967}{47606} a^{13} - \frac{549617}{142818} a^{12} + \frac{26899}{23803} a^{11} - \frac{324175}{71409} a^{10} + \frac{753937}{142818} a^{9} - \frac{346456}{71409} a^{8} - \frac{106321}{142818} a^{7} - \frac{17395}{71409} a^{6} + \frac{814865}{142818} a^{5} - \frac{345281}{47606} a^{4} - \frac{194975}{142818} a^{3} - \frac{34673}{142818} a^{2} + \frac{95847}{47606} a - \frac{6029}{23803} \)  (order $6$) 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:   $\frac{131732}{785499}a^{17}-\frac{294956}{785499}a^{16}+\frac{7037}{10986}a^{15}-\frac{457835}{523666}a^{14}+\frac{1958383}{1570998}a^{13}-\frac{2172241}{523666}a^{12}+\frac{497582}{785499}a^{11}-\frac{7582787}{1570998}a^{10}+\frac{7294097}{1570998}a^{9}-\frac{6308743}{1570998}a^{8}-\frac{2785624}{785499}a^{7}+\frac{116374}{60423}a^{6}+\frac{3492409}{523666}a^{5}-\frac{996653}{142818}a^{4}-\frac{1991395}{785499}a^{3}+\frac{834461}{523666}a^{2}+\frac{175514}{261833}a-\frac{49527}{523666}$, $\frac{560}{785499}a^{17}+\frac{9323}{142818}a^{16}+\frac{30983}{1570998}a^{15}+\frac{13417}{40282}a^{14}+\frac{7991}{40282}a^{13}+\frac{340625}{261833}a^{12}+\frac{819835}{1570998}a^{11}+\frac{3169981}{1570998}a^{10}-\frac{84949}{142818}a^{9}+\frac{1633343}{785499}a^{8}-\frac{215290}{261833}a^{7}+\frac{945657}{523666}a^{6}-\frac{779853}{523666}a^{5}+\frac{1770}{1831}a^{4}-\frac{168239}{1570998}a^{3}+\frac{1170337}{785499}a^{2}-\frac{2027635}{1570998}a-\frac{138400}{785499}$, $\frac{138400}{785499}a^{17}-\frac{137840}{785499}a^{16}+\frac{403251}{523666}a^{15}-\frac{573}{3662}a^{14}+\frac{116511}{47606}a^{13}-\frac{66047}{47606}a^{12}+\frac{894225}{261833}a^{11}-\frac{378415}{142818}a^{10}+\frac{10366781}{1570998}a^{9}-\frac{4256039}{1570998}a^{8}+\frac{1079743}{785499}a^{7}-\frac{1337870}{785499}a^{6}+\frac{11417771}{1570998}a^{5}-\frac{640469}{142818}a^{4}+\frac{114710}{261833}a^{3}-\frac{168239}{1570998}a^{2}+\frac{620779}{261833}a-\frac{1287037}{1570998}$, $\frac{130265}{523666}a^{17}+\frac{74992}{785499}a^{16}+\frac{56861}{71409}a^{15}+\frac{813652}{785499}a^{14}+\frac{1655091}{523666}a^{13}+\frac{2992549}{1570998}a^{12}+\frac{414400}{261833}a^{11}-\frac{339720}{261833}a^{10}+\frac{2212201}{1570998}a^{9}+\frac{2475851}{785499}a^{8}-\frac{1181269}{523666}a^{7}-\frac{1041855}{261833}a^{6}+\frac{2860431}{523666}a^{5}+\frac{602461}{142818}a^{4}-\frac{788313}{523666}a^{3}-\frac{814745}{523666}a^{2}-\frac{774733}{1570998}a-\frac{9719}{785499}$, $\frac{71515}{785499}a^{17}+\frac{99232}{785499}a^{16}+\frac{72660}{261833}a^{15}+\frac{630640}{785499}a^{14}+\frac{318849}{261833}a^{13}+\frac{1775770}{785499}a^{12}+\frac{289775}{785499}a^{11}+\frac{1133569}{785499}a^{10}-\frac{534795}{261833}a^{9}+\frac{831295}{261833}a^{8}-\frac{3040541}{785499}a^{7}-\frac{213685}{785499}a^{6}-\frac{1054145}{785499}a^{5}+\frac{221735}{71409}a^{4}+\frac{100990}{785499}a^{3}+\frac{29090}{261833}a^{2}-\frac{1144201}{785499}a+\frac{15245}{261833}$, $\frac{11201}{60423}a^{17}-\frac{5165}{523666}a^{16}+\frac{750743}{785499}a^{15}+\frac{279358}{785499}a^{14}+\frac{936820}{261833}a^{13}+\frac{1476997}{1570998}a^{12}+\frac{9095929}{1570998}a^{11}-\frac{1193626}{785499}a^{10}+\frac{5317330}{785499}a^{9}-\frac{183259}{120846}a^{8}+\frac{99692}{20141}a^{7}-\frac{1048871}{523666}a^{6}+\frac{877815}{261833}a^{5}+\frac{100597}{142818}a^{4}+\frac{9580381}{1570998}a^{3}-\frac{2476423}{1570998}a^{2}+\frac{120113}{142818}a+\frac{380617}{523666}$, $\frac{219748}{785499}a^{17}-\frac{215661}{523666}a^{16}+\frac{931243}{785499}a^{15}-\frac{479137}{785499}a^{14}+\frac{838769}{261833}a^{13}-\frac{5565673}{1570998}a^{12}+\frac{6304385}{1570998}a^{11}-\frac{3367196}{785499}a^{10}+\frac{7130768}{785499}a^{9}-\frac{6556985}{1570998}a^{8}-\frac{637882}{261833}a^{7}+\frac{1114115}{523666}a^{6}+\frac{2802212}{261833}a^{5}-\frac{1263811}{142818}a^{4}-\frac{2495917}{1570998}a^{3}+\frac{4347193}{1570998}a^{2}+\frac{3277271}{1570998}a-\frac{718093}{523666}$, $\frac{58948}{785499}a^{17}-\frac{22651}{71409}a^{16}+\frac{135688}{261833}a^{15}-\frac{795848}{785499}a^{14}+\frac{268977}{261833}a^{13}-\frac{922538}{261833}a^{12}+\frac{710962}{261833}a^{11}-\frac{3087947}{785499}a^{10}+\frac{404140}{71409}a^{9}-\frac{5125552}{785499}a^{8}+\frac{962792}{785499}a^{7}+\frac{1224341}{785499}a^{6}+\frac{1735535}{785499}a^{5}-\frac{575387}{71409}a^{4}+\frac{1218814}{785499}a^{3}+\frac{184774}{60423}a^{2}-\frac{10273}{261833}a-\frac{1002175}{785499}$ Copy content Toggle raw display
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:  \( 527.323608131 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 527.323608131 \cdot 1}{6\cdot\sqrt{43564677551979246963}}\cr\approx \mathstrut & 0.203225185150 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 4*x^16 - x^15 + 12*x^14 - 9*x^13 + 12*x^12 - 18*x^11 + 26*x^10 - 12*x^9 - 4*x^8 - 5*x^7 + 31*x^6 - 17*x^5 - 3*x^4 + 5*x^2 - 3*x + 1)
 
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 - x^17 + 4*x^16 - x^15 + 12*x^14 - 9*x^13 + 12*x^12 - 18*x^11 + 26*x^10 - 12*x^9 - 4*x^8 - 5*x^7 + 31*x^6 - 17*x^5 - 3*x^4 + 5*x^2 - 3*x + 1, 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 - x^17 + 4*x^16 - x^15 + 12*x^14 - 9*x^13 + 12*x^12 - 18*x^11 + 26*x^10 - 12*x^9 - 4*x^8 - 5*x^7 + 31*x^6 - 17*x^5 - 3*x^4 + 5*x^2 - 3*x + 1);
 
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 - x^17 + 4*x^16 - x^15 + 12*x^14 - 9*x^13 + 12*x^12 - 18*x^11 + 26*x^10 - 12*x^9 - 4*x^8 - 5*x^7 + 31*x^6 - 17*x^5 - 3*x^4 + 5*x^2 - 3*x + 1);
 
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

$C_3\times S_3$ (as 18T3):

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 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1083.1 x3, 3.3.361.1, 6.0.3518667.2, 6.0.9747.1 x2, 6.0.3518667.1, 9.3.1270238787.1 x3

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 6 sibling: 6.0.9747.1
Degree 9 sibling: 9.3.1270238787.1
Minimal sibling: 6.0.9747.1

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 ${\href{/padicField/5.6.0.1}{6} }^{3}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ R ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(19\) Copy content Toggle raw display 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$$[\ ]_{3}$
19.3.2.2$x^{3} + 19$$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}$