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Properties

Label 18.0.627...024.2
Degree $18$
Signature $[0, 9]$
Discriminant $-6.275\times 10^{41}$
Root discriminant \(209.94\)
Ramified primes $2,3,7$
Class number $5757156$ (GRH)
Class group [3, 9, 213228] (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 252*x^16 + 26460*x^14 + 1498224*x^12 + 49441392*x^10 + 958402368*x^8 + 10435936896*x^6 + 56923292160*x^4 + 119538913536*x^2 + 27678662144)
 
gp: K = bnfinit(y^18 + 252*y^16 + 26460*y^14 + 1498224*y^12 + 49441392*y^10 + 958402368*y^8 + 10435936896*y^6 + 56923292160*y^4 + 119538913536*y^2 + 27678662144, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 252*x^16 + 26460*x^14 + 1498224*x^12 + 49441392*x^10 + 958402368*x^8 + 10435936896*x^6 + 56923292160*x^4 + 119538913536*x^2 + 27678662144);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 252*x^16 + 26460*x^14 + 1498224*x^12 + 49441392*x^10 + 958402368*x^8 + 10435936896*x^6 + 56923292160*x^4 + 119538913536*x^2 + 27678662144)
 

\( x^{18} + 252 x^{16} + 26460 x^{14} + 1498224 x^{12} + 49441392 x^{10} + 958402368 x^{8} + \cdots + 27678662144 \) 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:   \(-627502832899903433922253495745464472961024\) \(\medspace = -\,2^{27}\cdot 3^{44}\cdot 7^{15}\) 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:  \(209.94\)
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:  $2^{3/2}3^{22/9}7^{5/6}\approx 209.9373740333372$
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:  \(\Q(\sqrt{-14}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1512=2^{3}\cdot 3^{3}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1512}(1,·)$, $\chi_{1512}(1033,·)$, $\chi_{1512}(685,·)$, $\chi_{1512}(529,·)$, $\chi_{1512}(1237,·)$, $\chi_{1512}(121,·)$, $\chi_{1512}(25,·)$, $\chi_{1512}(733,·)$, $\chi_{1512}(1189,·)$, $\chi_{1512}(229,·)$, $\chi_{1512}(625,·)$, $\chi_{1512}(1129,·)$, $\chi_{1512}(493,·)$, $\chi_{1512}(1501,·)$, $\chi_{1512}(1009,·)$, $\chi_{1512}(181,·)$, $\chi_{1512}(505,·)$, $\chi_{1512}(997,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{256}$

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{56}a^{6}$, $\frac{1}{56}a^{7}$, $\frac{1}{112}a^{8}$, $\frac{1}{44464}a^{9}+\frac{9}{3176}a^{7}+\frac{189}{1588}a^{5}-\frac{59}{397}a^{3}-\frac{89}{397}a$, $\frac{1}{28723744}a^{10}-\frac{10259}{14361872}a^{8}-\frac{63785}{7180936}a^{6}-\frac{33069}{512924}a^{4}-\frac{5404}{128231}a^{2}-\frac{71}{323}$, $\frac{1}{28723744}a^{11}+\frac{11}{2051696}a^{9}-\frac{13443}{1795234}a^{7}-\frac{1515}{256462}a^{5}+\frac{25927}{128231}a^{3}-\frac{50474}{128231}a$, $\frac{1}{402132416}a^{12}-\frac{4745}{14361872}a^{8}+\frac{28725}{3590468}a^{6}-\frac{31845}{512924}a^{4}-\frac{53191}{256462}a^{2}+\frac{135}{323}$, $\frac{1}{402132416}a^{13}+\frac{25}{3590468}a^{9}-\frac{393}{128231}a^{7}-\frac{13757}{512924}a^{5}+\frac{8127}{128231}a^{3}+\frac{7083}{128231}a$, $\frac{1}{804264832}a^{14}-\frac{5489}{3590468}a^{8}-\frac{559}{3590468}a^{6}-\frac{2713}{128231}a^{4}+\frac{3265}{13498}a^{2}-\frac{6}{323}$, $\frac{1}{804264832}a^{15}+\frac{1}{1795234}a^{9}-\frac{27927}{7180936}a^{7}+\frac{9238}{128231}a^{5}+\frac{1837}{13498}a^{3}-\frac{33713}{128231}a$, $\frac{1}{1608529664}a^{16}+\frac{455}{120688}a^{8}-\frac{13}{211204}a^{6}+\frac{2529}{30172}a^{4}+\frac{3103}{15086}a^{2}-\frac{78}{323}$, $\frac{1}{1608529664}a^{17}-\frac{1}{120688}a^{9}+\frac{2539}{422408}a^{7}+\frac{2681}{30172}a^{5}+\frac{2609}{15086}a^{3}+\frac{53983}{128231}a$ 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

$C_{3}\times C_{9}\times C_{213228}$, which has order $5757156$ (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:   $\frac{1}{25133276}a^{12}+\frac{6}{897617}a^{10}+\frac{54}{128231}a^{8}+\frac{85039}{7180936}a^{6}+\frac{32853}{256462}a^{4}+\frac{23121}{256462}a^{2}-\frac{167}{323}$, $\frac{55}{402132416}a^{12}+\frac{165}{7180936}a^{10}+\frac{1485}{1025848}a^{8}+\frac{37542}{897617}a^{6}+\frac{273951}{512924}a^{4}+\frac{292194}{128231}a^{2}+\frac{294}{323}$, $\frac{3}{1608529664}a^{16}+\frac{355}{804264832}a^{14}+\frac{17047}{402132416}a^{12}+\frac{7575}{3590468}a^{10}+\frac{825535}{14361872}a^{8}+\frac{2979685}{3590468}a^{6}+\frac{2897307}{512924}a^{4}+\frac{1864493}{128231}a^{2}+\frac{1888}{323}$, $\frac{1}{1608529664}a^{16}+\frac{113}{804264832}a^{14}+\frac{5297}{402132416}a^{12}+\frac{2367}{3590468}a^{10}+\frac{33815}{1795234}a^{8}+\frac{1088449}{3590468}a^{6}+\frac{639763}{256462}a^{4}+\frac{119345}{15086}a^{2}+\frac{702}{323}$, $\frac{3}{1608529664}a^{16}+\frac{317}{804264832}a^{14}+\frac{13527}{402132416}a^{12}+\frac{21401}{14361872}a^{10}+\frac{131723}{3590468}a^{8}+\frac{3578139}{7180936}a^{6}+\frac{1773189}{512924}a^{4}+\frac{2555079}{256462}a^{2}+\frac{1808}{323}$, $\frac{9}{402132416}a^{14}+\frac{445}{100533104}a^{12}+\frac{1229}{3590468}a^{10}+\frac{186171}{14361872}a^{8}+\frac{1751473}{7180936}a^{6}+\frac{1012153}{512924}a^{4}+\frac{853909}{256462}a^{2}-\frac{1804}{323}$, $\frac{3}{1608529664}a^{16}+\frac{3}{7180936}a^{14}+\frac{15233}{402132416}a^{12}+\frac{50997}{28723744}a^{10}+\frac{326633}{7180936}a^{8}+\frac{4447211}{7180936}a^{6}+\frac{2073947}{512924}a^{4}+\frac{1329714}{128231}a^{2}+\frac{1906}{323}$, $\frac{1}{804264832}a^{16}+\frac{1}{3590468}a^{14}+\frac{10247}{402132416}a^{12}+\frac{5001}{4103392}a^{10}+\frac{27257}{844816}a^{8}+\frac{172931}{377944}a^{6}+\frac{1557233}{512924}a^{4}+\frac{44178}{6749}a^{2}+\frac{362}{323}$ 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:  \( 10392888.21418944 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 10392888.21418944 \cdot 5757156}{2\cdot\sqrt{627502832899903433922253495745464472961024}}\cr\approx \mathstrut & 0.576401666480501 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 + 252*x^16 + 26460*x^14 + 1498224*x^12 + 49441392*x^10 + 958402368*x^8 + 10435936896*x^6 + 56923292160*x^4 + 119538913536*x^2 + 27678662144)
 
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 + 252*x^16 + 26460*x^14 + 1498224*x^12 + 49441392*x^10 + 958402368*x^8 + 10435936896*x^6 + 56923292160*x^4 + 119538913536*x^2 + 27678662144, 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 + 252*x^16 + 26460*x^14 + 1498224*x^12 + 49441392*x^10 + 958402368*x^8 + 10435936896*x^6 + 56923292160*x^4 + 119538913536*x^2 + 27678662144);
 
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 + 252*x^16 + 26460*x^14 + 1498224*x^12 + 49441392*x^10 + 958402368*x^8 + 10435936896*x^6 + 56923292160*x^4 + 119538913536*x^2 + 27678662144);
 
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

$C_{18}$ (as 18T1):

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 cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-14}) \), \(\Q(\zeta_{9})^+\), 6.0.1152216576.2, 9.9.3691950281939241.2

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]
 

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 ${\href{/padicField/5.9.0.1}{9} }^{2}$ R $18$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.1.0.1}{1} }^{18}$ ${\href{/padicField/23.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ $18$ $18$ $18$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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 2.18.27.119$x^{18} + 70 x^{16} + 1024 x^{15} - 9632 x^{14} + 37696 x^{13} + 414592 x^{12} - 7305728 x^{11} + 60591136 x^{10} - 292750080 x^{9} + 723623360 x^{8} - 693690368 x^{7} + 2330844160 x^{6} - 20528915456 x^{5} + 72224523264 x^{4} - 130591481856 x^{3} + 133128467712 x^{2} - 74944811008 x + 19148434944$$2$$9$$27$$C_{18}$$[3]^{9}$
\(3\) Copy content Toggle raw display 3.9.22.6$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.6$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$$9$$1$$22$$C_9$$[2, 3]$
\(7\) Copy content Toggle raw display 7.18.15.6$x^{18} + 189 x^{12} + 6027 x^{6} + 95011$$6$$3$$15$$C_{18}$$[\ ]_{6}^{3}$