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Properties

Label 18.0.274...447.1
Degree $18$
Signature $[0, 9]$
Discriminant $-2.746\times 10^{18}$
Root discriminant \(10.58\)
Ramified primes $23,2647$
Class number $1$
Class group trivial
Galois group $S_3\wr S_3$ (as 18T314)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 12*x^16 - 17*x^15 + 12*x^14 - 2*x^13 + 8*x^12 - 46*x^11 + 111*x^10 - 168*x^9 + 188*x^8 - 173*x^7 + 137*x^6 - 102*x^5 + 70*x^4 - 40*x^3 + 19*x^2 - 5*x + 1)
 
gp: K = bnfinit(y^18 - 5*y^17 + 12*y^16 - 17*y^15 + 12*y^14 - 2*y^13 + 8*y^12 - 46*y^11 + 111*y^10 - 168*y^9 + 188*y^8 - 173*y^7 + 137*y^6 - 102*y^5 + 70*y^4 - 40*y^3 + 19*y^2 - 5*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 5*x^17 + 12*x^16 - 17*x^15 + 12*x^14 - 2*x^13 + 8*x^12 - 46*x^11 + 111*x^10 - 168*x^9 + 188*x^8 - 173*x^7 + 137*x^6 - 102*x^5 + 70*x^4 - 40*x^3 + 19*x^2 - 5*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 5*x^17 + 12*x^16 - 17*x^15 + 12*x^14 - 2*x^13 + 8*x^12 - 46*x^11 + 111*x^10 - 168*x^9 + 188*x^8 - 173*x^7 + 137*x^6 - 102*x^5 + 70*x^4 - 40*x^3 + 19*x^2 - 5*x + 1)
 

\( x^{18} - 5 x^{17} + 12 x^{16} - 17 x^{15} + 12 x^{14} - 2 x^{13} + 8 x^{12} - 46 x^{11} + 111 x^{10} + \cdots + 1 \) 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:   \(-2745546730527991447\) \(\medspace = -\,23^{6}\cdot 2647^{3}\) 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:  \(10.58\)
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:  $23^{1/2}2647^{1/2}\approx 246.74075463935827$
Ramified primes:   \(23\), \(2647\) 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:  \(\Q(\sqrt{-2647}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{107897}a^{17}+\frac{40872}{107897}a^{16}+\frac{47608}{107897}a^{15}+\frac{41907}{107897}a^{14}-\frac{48218}{107897}a^{13}-\frac{52689}{107897}a^{12}-\frac{36228}{107897}a^{11}-\frac{5677}{107897}a^{10}+\frac{27829}{107897}a^{9}+\frac{7794}{107897}a^{8}-\frac{24315}{107897}a^{7}+\frac{22736}{107897}a^{6}-\frac{45149}{107897}a^{5}+\frac{22410}{107897}a^{4}+\frac{8110}{107897}a^{3}+\frac{52846}{107897}a^{2}-\frac{19876}{107897}a-\frac{6847}{107897}$ 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:   \( -1 \)  (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:   $\frac{22226}{107897}a^{17}-\frac{179565}{107897}a^{16}+\frac{636911}{107897}a^{15}-\frac{1128789}{107897}a^{14}+\frac{1018706}{107897}a^{13}-\frac{59573}{107897}a^{12}-\frac{399805}{107897}a^{11}-\frac{1879658}{107897}a^{10}+\frac{6211879}{107897}a^{9}-\frac{10303353}{107897}a^{8}+\frac{11791656}{107897}a^{7}-\frac{9985736}{107897}a^{6}+\frac{7944804}{107897}a^{5}-\frac{5794330}{107897}a^{4}+\frac{3841265}{107897}a^{3}-\frac{2493177}{107897}a^{2}+\frac{721621}{107897}a-\frac{262446}{107897}$, $\frac{68176}{107897}a^{17}-\frac{274244}{107897}a^{16}+\frac{612836}{107897}a^{15}-\frac{816207}{107897}a^{14}+\frac{527016}{107897}a^{13}-\frac{126237}{107897}a^{12}+\frac{421687}{107897}a^{11}-\frac{2382347}{107897}a^{10}+\frac{5727597}{107897}a^{9}-\frac{8444947}{107897}a^{8}+\frac{9309210}{107897}a^{7}-\frac{7875247}{107897}a^{6}+\frac{5725933}{107897}a^{5}-\frac{4205343}{107897}a^{4}+\frac{2632660}{107897}a^{3}-\frac{1578286}{107897}a^{2}+\frac{443835}{107897}a+\frac{69247}{107897}$, $\frac{6437}{107897}a^{17}-\frac{175616}{107897}a^{16}+\frac{672598}{107897}a^{15}-\frac{1281905}{107897}a^{14}+\frac{1227270}{107897}a^{13}-\frac{38822}{107897}a^{12}-\frac{573704}{107897}a^{11}-\frac{1692118}{107897}a^{10}+\frac{6715867}{107897}a^{9}-\frac{11655003}{107897}a^{8}+\frac{13098429}{107897}a^{7}-\frac{11393782}{107897}a^{6}+\frac{8790062}{107897}a^{5}-\frac{6478939}{107897}a^{4}+\frac{4729390}{107897}a^{3}-\frac{2619067}{107897}a^{2}+\frac{995103}{107897}a-\frac{160060}{107897}$, $\frac{70645}{107897}a^{17}-\frac{244971}{107897}a^{16}+\frac{225567}{107897}a^{15}+\frac{257923}{107897}a^{14}-\frac{915496}{107897}a^{13}+\frac{447889}{107897}a^{12}+\frac{1500338}{107897}a^{11}-\frac{2480147}{107897}a^{10}+\frac{1283232}{107897}a^{9}+\frac{1411400}{107897}a^{8}-\frac{3573536}{107897}a^{7}+\frac{3051094}{107897}a^{6}-\frac{2165828}{107897}a^{5}+\frac{1708121}{107897}a^{4}-\frac{1296884}{107897}a^{3}+\frac{1256337}{107897}a^{2}-\frac{292153}{107897}a-\frac{4064}{107897}$, $\frac{48842}{107897}a^{17}-\frac{255864}{107897}a^{16}+\frac{521174}{107897}a^{15}-\frac{523881}{107897}a^{14}+\frac{4263}{107897}a^{13}+\frac{446797}{107897}a^{12}+\frac{494412}{107897}a^{11}-\frac{2570272}{107897}a^{10}+\frac{4685080}{107897}a^{9}-\frac{5380918}{107897}a^{8}+\frac{4021238}{107897}a^{7}-\frac{2809534}{107897}a^{6}+\frac{1865677}{107897}a^{5}-\frac{1252712}{107897}a^{4}+\frac{881909}{107897}a^{3}+\frac{100195}{107897}a^{2}-\frac{142180}{107897}a+\frac{167423}{107897}$, $\frac{61558}{107897}a^{17}-\frac{159464}{107897}a^{16}+\frac{62847}{107897}a^{15}+\frac{325424}{107897}a^{14}-\frac{712453}{107897}a^{13}+\frac{170152}{107897}a^{12}+\frac{1078839}{107897}a^{11}-\frac{1281147}{107897}a^{10}+\frac{232707}{107897}a^{9}+\frac{1907239}{107897}a^{8}-\frac{3488290}{107897}a^{7}+\frac{3071817}{107897}a^{6}-\frac{2876538}{107897}a^{5}+\frac{2317472}{107897}a^{4}-\frac{1730391}{107897}a^{3}+\frac{1402179}{107897}a^{2}-\frac{298519}{107897}a+\frac{173850}{107897}$, $\frac{115087}{107897}a^{17}-\frac{581233}{107897}a^{16}+\frac{1239103}{107897}a^{15}-\frac{1339755}{107897}a^{14}+\frac{201435}{107897}a^{13}+\frac{855633}{107897}a^{12}+\frac{1063008}{107897}a^{11}-\frac{5966899}{107897}a^{10}+\frac{11162863}{107897}a^{9}-\frac{13231014}{107897}a^{8}+\frac{11297475}{107897}a^{7}-\frac{8623875}{107897}a^{6}+\frac{6082995}{107897}a^{5}-\frac{4170304}{107897}a^{4}+\frac{2636048}{107897}a^{3}-\frac{697876}{107897}a^{2}+\frac{162982}{107897}a+\frac{78999}{107897}$, $\frac{83016}{107897}a^{17}-\frac{438595}{107897}a^{16}+\frac{1037588}{107897}a^{15}-\frac{1374120}{107897}a^{14}+\frac{760594}{107897}a^{13}+\frac{222253}{107897}a^{12}+\frac{556815}{107897}a^{11}-\frac{4195719}{107897}a^{10}+\frac{9564533}{107897}a^{9}-\frac{13518730}{107897}a^{8}+\frac{13813852}{107897}a^{7}-\frac{11643321}{107897}a^{6}+\frac{8776259}{107897}a^{5}-\frac{6337437}{107897}a^{4}+\frac{4082566}{107897}a^{3}-\frac{1970630}{107897}a^{2}+\frac{690187}{107897}a-\frac{9156}{107897}$ 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:  \( 35.6314081744 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 35.6314081744 \cdot 1}{2\cdot\sqrt{2745546730527991447}}\cr\approx \mathstrut & 0.164099548377 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 12*x^16 - 17*x^15 + 12*x^14 - 2*x^13 + 8*x^12 - 46*x^11 + 111*x^10 - 168*x^9 + 188*x^8 - 173*x^7 + 137*x^6 - 102*x^5 + 70*x^4 - 40*x^3 + 19*x^2 - 5*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 - 5*x^17 + 12*x^16 - 17*x^15 + 12*x^14 - 2*x^13 + 8*x^12 - 46*x^11 + 111*x^10 - 168*x^9 + 188*x^8 - 173*x^7 + 137*x^6 - 102*x^5 + 70*x^4 - 40*x^3 + 19*x^2 - 5*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 - 5*x^17 + 12*x^16 - 17*x^15 + 12*x^14 - 2*x^13 + 8*x^12 - 46*x^11 + 111*x^10 - 168*x^9 + 188*x^8 - 173*x^7 + 137*x^6 - 102*x^5 + 70*x^4 - 40*x^3 + 19*x^2 - 5*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 - 5*x^17 + 12*x^16 - 17*x^15 + 12*x^14 - 2*x^13 + 8*x^12 - 46*x^11 + 111*x^10 - 168*x^9 + 188*x^8 - 173*x^7 + 137*x^6 - 102*x^5 + 70*x^4 - 40*x^3 + 19*x^2 - 5*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

$S_3\wr S_3$ (as 18T314):

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 1296
The 22 conjugacy class representatives for $S_3\wr S_3$
Character table for $S_3\wr S_3$

Intermediate fields

3.1.23.1, 6.0.1400263.1, 9.1.32206049.1 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 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 27 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 9.1.32206049.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.9.0.1}{9} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }^{3}$ ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{9}$ R ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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
\(23\) Copy content Toggle raw display 23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(2647\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$