Properties

Label 18.0.520...152.2
Degree $18$
Signature $[0, 9]$
Discriminant $-5.210\times 10^{29}$
Root discriminant \(44.76\)
Ramified primes $2,3,7$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 + 87*x^12 - 848*x^9 + 7392*x^6 - 18816*x^3 + 21952)
 
gp: K = bnfinit(y^18 - 12*y^15 + 87*y^12 - 848*y^9 + 7392*y^6 - 18816*y^3 + 21952, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 12*x^15 + 87*x^12 - 848*x^9 + 7392*x^6 - 18816*x^3 + 21952);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 12*x^15 + 87*x^12 - 848*x^9 + 7392*x^6 - 18816*x^3 + 21952)
 

\( x^{18} - 12x^{15} + 87x^{12} - 848x^{9} + 7392x^{6} - 18816x^{3} + 21952 \) 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:   \(-520986863358984384396363313152\) \(\medspace = -\,2^{12}\cdot 3^{37}\cdot 7^{10}\) 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:  \(44.76\)
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:  $2\cdot 3^{121/54}7^{5/6}\approx 118.68160094629725$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{3}+\frac{1}{3}$, $\frac{1}{12}a^{10}+\frac{1}{4}a^{4}-\frac{1}{3}a$, $\frac{1}{36}a^{11}+\frac{1}{36}a^{10}-\frac{1}{18}a^{9}-\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}+\frac{1}{12}a^{5}-\frac{5}{12}a^{4}+\frac{1}{3}a^{3}-\frac{5}{18}a^{2}+\frac{2}{9}a-\frac{4}{9}$, $\frac{1}{1008}a^{12}-\frac{5}{126}a^{9}-\frac{9}{112}a^{6}-\frac{37}{252}a^{3}-\frac{1}{18}$, $\frac{1}{2016}a^{13}-\frac{5}{252}a^{10}-\frac{9}{224}a^{7}+\frac{215}{504}a^{4}-\frac{1}{36}a$, $\frac{1}{2016}a^{14}+\frac{1}{126}a^{11}+\frac{1}{36}a^{10}-\frac{1}{18}a^{9}-\frac{139}{672}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}+\frac{5}{504}a^{5}-\frac{5}{12}a^{4}+\frac{1}{3}a^{3}+\frac{7}{36}a^{2}+\frac{2}{9}a-\frac{4}{9}$, $\frac{1}{12672576}a^{15}+\frac{55}{132006}a^{12}-\frac{747457}{12672576}a^{9}-\frac{248887}{3168144}a^{6}+\frac{10565}{75432}a^{3}+\frac{1015}{4041}$, $\frac{1}{12672576}a^{16}-\frac{503}{6336288}a^{13}-\frac{55113}{1408064}a^{10}-\frac{243191}{6336288}a^{7}-\frac{8105}{28287}a^{4}+\frac{501}{1796}a$, $\frac{1}{177416064}a^{17}-\frac{2483}{22177008}a^{14}-\frac{1149761}{177416064}a^{11}+\frac{1}{36}a^{10}-\frac{1}{18}a^{9}-\frac{5547985}{44354016}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{45247}{452592}a^{5}-\frac{5}{12}a^{4}+\frac{1}{3}a^{3}+\frac{7018}{28287}a^{2}+\frac{2}{9}a-\frac{4}{9}$ 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

$C_{3}$, 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:   \( -\frac{37}{258624} a^{15} + \frac{101}{64656} a^{12} - \frac{2491}{258624} a^{9} + \frac{3731}{32328} a^{6} - \frac{30919}{32328} a^{3} + \frac{15329}{8082} \)  (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:   $\frac{943}{12672576}a^{15}-\frac{3077}{1584072}a^{12}+\frac{28411}{4224192}a^{9}-\frac{314359}{3168144}a^{6}+\frac{229241}{226296}a^{3}-\frac{1837}{1347}$, $\frac{1153}{12672576}a^{17}+\frac{51}{1408064}a^{16}+\frac{117}{1408064}a^{15}-\frac{7933}{6336288}a^{14}-\frac{719}{3168144}a^{13}-\frac{461}{264012}a^{12}+\frac{118399}{12672576}a^{11}+\frac{32261}{12672576}a^{10}+\frac{45377}{4224192}a^{9}-\frac{414497}{6336288}a^{8}-\frac{8759}{176008}a^{7}-\frac{15599}{352016}a^{6}+\frac{14789}{28287}a^{5}+\frac{84817}{226296}a^{4}+\frac{13093}{75432}a^{3}-\frac{14911}{16164}a^{2}-\frac{5293}{8082}a-\frac{1139}{1347}$, $\frac{299}{25345152}a^{17}-\frac{269}{6336288}a^{16}+\frac{117}{1408064}a^{15}+\frac{1805}{6336288}a^{14}+\frac{1153}{2112096}a^{13}-\frac{461}{264012}a^{12}-\frac{110347}{25345152}a^{11}-\frac{35779}{6336288}a^{10}+\frac{45377}{4224192}a^{9}+\frac{87803}{3168144}a^{8}+\frac{405401}{6336288}a^{7}-\frac{15599}{352016}a^{6}-\frac{60853}{452592}a^{5}-\frac{27019}{75432}a^{4}+\frac{13093}{75432}a^{3}+\frac{11147}{16164}a^{2}+\frac{4595}{16164}a-\frac{1139}{1347}$, $\frac{71}{8448384}a^{17}-\frac{1019}{12672576}a^{16}-\frac{55}{528024}a^{15}-\frac{95}{352016}a^{14}+\frac{363}{352016}a^{13}+\frac{655}{1056048}a^{12}+\frac{34681}{8448384}a^{11}-\frac{53653}{12672576}a^{10}-\frac{305}{528024}a^{9}-\frac{1031}{2112096}a^{8}+\frac{124597}{1584072}a^{7}+\frac{105593}{1056048}a^{6}+\frac{1965}{7184}a^{5}-\frac{11545}{25144}a^{4}+\frac{5009}{37716}a^{3}-\frac{1010}{1347}a^{2}+\frac{4463}{8082}a-\frac{139}{2694}$, $\frac{1753}{5544252}a^{17}-\frac{835}{2112096}a^{16}-\frac{829}{3168144}a^{15}-\frac{8005}{2772126}a^{14}+\frac{27631}{6336288}a^{13}+\frac{2111}{1584072}a^{12}+\frac{112583}{5544252}a^{11}-\frac{194471}{6336288}a^{10}-\frac{32027}{3168144}a^{9}-\frac{613331}{2772126}a^{8}+\frac{222085}{704032}a^{7}+\frac{239615}{1584072}a^{6}+\frac{50945}{28287}a^{5}-\frac{648313}{226296}a^{4}-\frac{53555}{56574}a^{3}-\frac{23024}{28287}a^{2}+\frac{90247}{16164}a-\frac{10363}{4041}$, $\frac{7897}{22177008}a^{17}-\frac{6157}{12672576}a^{16}+\frac{829}{3168144}a^{15}-\frac{43441}{11088504}a^{14}+\frac{3267}{704032}a^{13}-\frac{2111}{1584072}a^{12}+\frac{65567}{2464112}a^{11}-\frac{138625}{4224192}a^{10}+\frac{32027}{3168144}a^{9}-\frac{3054461}{11088504}a^{8}+\frac{2264747}{6336288}a^{7}-\frac{239615}{1584072}a^{6}+\frac{70487}{28287}a^{5}-\frac{27031}{9429}a^{4}+\frac{53555}{56574}a^{3}-\frac{39187}{9429}a^{2}+\frac{4463}{1796}a+\frac{10363}{4041}$, $\frac{92011}{88708032}a^{17}+\frac{2263}{1584072}a^{16}+\frac{36073}{12672576}a^{15}-\frac{437735}{44354016}a^{14}-\frac{1455}{88004}a^{13}-\frac{5113}{198009}a^{12}+\frac{5989349}{88708032}a^{11}+\frac{49709}{528024}a^{10}+\frac{2113271}{12672576}a^{9}-\frac{29498243}{44354016}a^{8}-\frac{856829}{792036}a^{7}-\frac{6444967}{3168144}a^{6}+\frac{684787}{113148}a^{5}+\frac{51389}{5388}a^{4}+\frac{3267059}{226296}a^{3}-\frac{431237}{113148}a^{2}-\frac{12344}{1347}a-\frac{70103}{4041}$, $\frac{70937}{88708032}a^{17}+\frac{6203}{6336288}a^{16}+\frac{223}{176008}a^{15}-\frac{84569}{11088504}a^{14}-\frac{57937}{6336288}a^{13}-\frac{3349}{264012}a^{12}+\frac{4327271}{88708032}a^{11}+\frac{354413}{6336288}a^{10}+\frac{40283}{528024}a^{9}-\frac{12412349}{22177008}a^{8}-\frac{4320611}{6336288}a^{7}-\frac{83989}{88004}a^{6}+\frac{972473}{226296}a^{5}+\frac{1117573}{226296}a^{4}+\frac{117947}{18858}a^{3}-\frac{134387}{28287}a^{2}-\frac{90709}{16164}a-\frac{13192}{1347}$ 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:  \( 174512587.28025168 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 174512587.28025168 \cdot 3}{6\cdot\sqrt{520986863358984384396363313152}}\cr\approx \mathstrut & 1.84502472510635 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 + 87*x^12 - 848*x^9 + 7392*x^6 - 18816*x^3 + 21952)
 
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 - 12*x^15 + 87*x^12 - 848*x^9 + 7392*x^6 - 18816*x^3 + 21952, 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 - 12*x^15 + 87*x^12 - 848*x^9 + 7392*x^6 - 18816*x^3 + 21952);
 
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 - 12*x^15 + 87*x^12 - 848*x^9 + 7392*x^6 - 18816*x^3 + 21952);
 
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

$C_3\times S_3^2$ (as 18T46):

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 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.756.1, 6.0.8680203.5, 6.0.1714608.3

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 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.0.740989189386333425664.1

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 ${\href{/padicField/5.6.0.1}{6} }^{3}$ R ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} + x + 1$$1$$2$$0$$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}$
\(3\) Copy content Toggle raw display Deg $18$$18$$1$$37$
\(7\) Copy content Toggle raw display 7.3.2.2$x^{3} + 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.5.2$x^{6} + 42$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.3.1$x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$