Properties

Label 18.0.817...000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-8.174\times 10^{18}$
Root discriminant \(11.24\)
Ramified primes $2,5,83$
Class number $1$
Class group trivial
Galois group $C_6^2:D_6$ (as 18T156)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{18} + x^{16} - 6x^{14} - 5x^{12} + 2x^{10} + 7x^{8} + 18x^{6} + 7x^{4} - 3x^{2} + 1 \) 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:   \(-8173509334225000000\) \(\medspace = -\,2^{6}\cdot 5^{8}\cdot 83^{6}\) 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:  \(11.24\)
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:  $2\cdot 5^{2/3}83^{1/2}\approx 53.27813877645612$
Ramified primes:   \(2\), \(5\), \(83\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6}a^{12}+\frac{1}{6}a^{10}-\frac{1}{3}a^{8}-\frac{1}{2}a^{7}-\frac{1}{3}a^{6}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}-\frac{1}{3}a^{2}-\frac{1}{2}a+\frac{1}{6}$, $\frac{1}{6}a^{13}+\frac{1}{6}a^{11}-\frac{1}{3}a^{9}-\frac{1}{2}a^{8}-\frac{1}{3}a^{7}-\frac{1}{2}a^{6}-\frac{1}{6}a^{5}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a$, $\frac{1}{6}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{6}$, $\frac{1}{6}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}+\frac{1}{6}a^{7}-\frac{1}{2}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}a$, $\frac{1}{582}a^{16}+\frac{7}{194}a^{14}+\frac{13}{291}a^{12}-\frac{1}{2}a^{11}+\frac{112}{291}a^{10}-\frac{1}{2}a^{9}+\frac{39}{194}a^{8}-\frac{136}{291}a^{6}-\frac{1}{2}a^{5}-\frac{281}{582}a^{4}-\frac{139}{291}a^{2}-\frac{17}{291}$, $\frac{1}{582}a^{17}+\frac{7}{194}a^{15}+\frac{13}{291}a^{13}+\frac{112}{291}a^{11}+\frac{39}{194}a^{9}+\frac{19}{582}a^{7}-\frac{1}{2}a^{6}+\frac{5}{291}a^{5}-\frac{1}{2}a^{4}-\frac{139}{291}a^{3}+\frac{257}{582}a-\frac{1}{2}$ 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:   $\frac{81}{194}a^{17}-\frac{91}{582}a^{16}+\frac{253}{582}a^{15}-\frac{34}{291}a^{14}-\frac{721}{291}a^{13}+\frac{272}{291}a^{12}-\frac{1343}{582}a^{11}+\frac{277}{582}a^{10}+\frac{301}{582}a^{9}-\frac{57}{194}a^{8}+\frac{1901}{582}a^{7}-\frac{78}{97}a^{6}+\frac{2282}{291}a^{5}-\frac{1589}{582}a^{4}+\frac{2383}{582}a^{3}-\frac{19}{582}a^{2}-\frac{19}{97}a+\frac{63}{97}$, $\frac{49}{291}a^{17}-\frac{41}{194}a^{16}+\frac{59}{291}a^{15}-\frac{61}{582}a^{14}-\frac{278}{291}a^{13}+\frac{341}{291}a^{12}-\frac{649}{582}a^{11}+\frac{95}{291}a^{10}-\frac{77}{582}a^{9}-\frac{35}{582}a^{8}+\frac{446}{291}a^{7}-\frac{147}{97}a^{6}+\frac{747}{194}a^{5}-\frac{507}{194}a^{4}+\frac{180}{97}a^{3}+\frac{122}{291}a^{2}+\frac{80}{291}a+\frac{151}{291}$, $\frac{45}{194}a^{17}+\frac{11}{291}a^{16}+\frac{36}{97}a^{15}+\frac{57}{194}a^{14}-\frac{379}{291}a^{13}-\frac{5}{291}a^{12}-\frac{1091}{582}a^{11}-\frac{446}{291}a^{10}+\frac{89}{291}a^{9}-\frac{56}{97}a^{8}+\frac{458}{291}a^{7}+\frac{127}{582}a^{6}+\frac{1354}{291}a^{5}+\frac{1093}{582}a^{4}+\frac{926}{291}a^{3}+\frac{1016}{291}a^{2}-\frac{419}{582}a-\frac{83}{291}$, $\frac{55}{291}a^{17}-\frac{24}{97}a^{16}+\frac{88}{291}a^{15}-\frac{211}{582}a^{14}-\frac{535}{582}a^{13}+\frac{815}{582}a^{12}-\frac{871}{582}a^{11}+\frac{556}{291}a^{10}-\frac{419}{582}a^{9}-\frac{67}{582}a^{8}+\frac{635}{582}a^{7}-\frac{592}{291}a^{6}+\frac{2555}{582}a^{5}-\frac{1496}{291}a^{4}+\frac{703}{194}a^{3}-\frac{839}{291}a^{2}+\frac{273}{194}a+\frac{40}{97}$, $\frac{45}{194}a^{17}-\frac{11}{291}a^{16}+\frac{36}{97}a^{15}-\frac{57}{194}a^{14}-\frac{379}{291}a^{13}+\frac{5}{291}a^{12}-\frac{1091}{582}a^{11}+\frac{446}{291}a^{10}+\frac{89}{291}a^{9}+\frac{56}{97}a^{8}+\frac{458}{291}a^{7}-\frac{127}{582}a^{6}+\frac{1354}{291}a^{5}-\frac{1093}{582}a^{4}+\frac{926}{291}a^{3}-\frac{1016}{291}a^{2}-\frac{419}{582}a+\frac{83}{291}$, $\frac{91}{291}a^{17}+\frac{68}{291}a^{15}-\frac{544}{291}a^{13}-\frac{277}{291}a^{11}+\frac{57}{97}a^{9}+\frac{156}{97}a^{7}+\frac{1589}{291}a^{5}+\frac{310}{291}a^{3}-\frac{126}{97}a$, $\frac{43}{291}a^{17}+\frac{10}{97}a^{15}-\frac{80}{97}a^{13}-\frac{55}{97}a^{11}-\frac{110}{291}a^{9}+\frac{332}{291}a^{7}+\frac{305}{97}a^{5}+\frac{656}{291}a^{3}+\frac{127}{97}a$, $a$ 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:  \( 72.3431657313 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 72.3431657313 \cdot 1}{2\cdot\sqrt{8173509334225000000}}\cr\approx \mathstrut & 0.193099899447 \end{aligned}\]

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

$C_6^2:D_6$ (as 18T156):

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 432
The 38 conjugacy class representatives for $C_6^2:D_6$
Character table for $C_6^2:D_6$ is not computed

Intermediate fields

3.1.83.1, 6.0.27556.1, 9.3.357366875.1

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 18 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}$ R ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.3.0.1}{3} }^{6}$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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
\(2\) Copy content Toggle raw display 2.6.6.5$x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$$2$$3$$6$$C_6$$[2]^{3}$
2.12.0.1$x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(5\) Copy content Toggle raw display 5.6.0.1$x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.8.2$x^{12} + 100 x^{6} - 500 x^{3} + 1250$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
\(83\) Copy content Toggle raw display 83.6.0.1$x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
83.6.3.2$x^{6} + 255 x^{4} + 162 x^{3} + 20676 x^{2} - 39852 x + 537761$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
83.6.3.2$x^{6} + 255 x^{4} + 162 x^{3} + 20676 x^{2} - 39852 x + 537761$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$