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Properties

Label 18.0.137...375.2
Degree $18$
Signature $[0, 9]$
Discriminant $-1.374\times 10^{28}$
Root discriminant \(36.58\)
Ramified primes $3,5$
Class number $4$ (GRH)
Class group [2, 2] (GRH)
Galois group $C_6:S_3$ (as 18T12)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^15 + 30*x^12 - 250*x^9 + 465*x^6 + 237*x^3 + 64)
 
gp: K = bnfinit(y^18 - 3*y^15 + 30*y^12 - 250*y^9 + 465*y^6 + 237*y^3 + 64, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^15 + 30*x^12 - 250*x^9 + 465*x^6 + 237*x^3 + 64);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^15 + 30*x^12 - 250*x^9 + 465*x^6 + 237*x^3 + 64)
 

\( x^{18} - 3x^{15} + 30x^{12} - 250x^{9} + 465x^{6} + 237x^{3} + 64 \) 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:   \(-13741574276458659759521484375\) \(\medspace = -\,3^{37}\cdot 5^{15}\) 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:  \(36.58\)
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:  $3^{37/18}5^{5/6}\approx 36.57836164674376$
Ramified primes:   \(3\), \(5\) 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:  \(\Q(\sqrt{-15}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{12}a^{9}+\frac{1}{4}a^{3}+\frac{1}{3}$, $\frac{1}{12}a^{10}-\frac{1}{4}a^{4}-\frac{1}{6}a$, $\frac{1}{48}a^{11}-\frac{1}{16}a^{8}+\frac{1}{16}a^{5}+\frac{7}{48}a^{2}$, $\frac{1}{48}a^{12}+\frac{1}{48}a^{9}+\frac{1}{16}a^{6}+\frac{19}{48}a^{3}+\frac{1}{3}$, $\frac{1}{48}a^{13}+\frac{1}{48}a^{10}+\frac{1}{16}a^{7}-\frac{5}{48}a^{4}-\frac{1}{6}a$, $\frac{1}{144}a^{14}-\frac{1}{144}a^{13}+\frac{1}{144}a^{12}+\frac{1}{144}a^{11}-\frac{1}{144}a^{10}+\frac{1}{144}a^{9}-\frac{1}{16}a^{8}+\frac{1}{16}a^{7}+\frac{3}{16}a^{6}-\frac{29}{144}a^{5}+\frac{29}{144}a^{4}+\frac{43}{144}a^{3}-\frac{5}{36}a^{2}+\frac{5}{36}a+\frac{1}{9}$, $\frac{1}{6624}a^{15}-\frac{2}{207}a^{12}-\frac{103}{3312}a^{9}+\frac{14}{207}a^{6}+\frac{461}{6624}a^{3}+\frac{86}{207}$, $\frac{1}{6624}a^{16}-\frac{2}{207}a^{13}-\frac{103}{3312}a^{10}+\frac{14}{207}a^{7}+\frac{461}{6624}a^{4}+\frac{86}{207}a$, $\frac{1}{6624}a^{17}-\frac{1}{368}a^{14}-\frac{1}{144}a^{13}+\frac{1}{144}a^{12}-\frac{11}{3312}a^{11}-\frac{1}{144}a^{10}+\frac{1}{144}a^{9}-\frac{95}{1656}a^{8}+\frac{1}{16}a^{7}+\frac{3}{16}a^{6}-\frac{51}{736}a^{5}+\frac{29}{144}a^{4}+\frac{43}{144}a^{3}+\frac{1399}{3312}a^{2}+\frac{5}{36}a+\frac{1}{9}$ 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

$C_{2}\times C_{2}$, which has order $4$ (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:   $\frac{1}{207}a^{17}-\frac{29}{1656}a^{14}+\frac{499}{3312}a^{11}-\frac{4217}{3312}a^{8}+\frac{9377}{3312}a^{5}+\frac{631}{3312}a^{2}$, $\frac{103}{1656}a^{17}-\frac{695}{3312}a^{14}+\frac{401}{207}a^{11}-\frac{26927}{1656}a^{8}+\frac{57557}{1656}a^{5}+\frac{11117}{3312}a^{2}$, $\frac{1}{2208}a^{17}-\frac{79}{6624}a^{16}-\frac{1}{1656}a^{15}-\frac{3}{368}a^{14}+\frac{113}{3312}a^{13}-\frac{5}{1656}a^{12}+\frac{35}{1104}a^{11}-\frac{589}{1656}a^{10}-\frac{1}{1656}a^{9}-\frac{41}{138}a^{8}+\frac{9835}{3312}a^{7}+\frac{173}{1656}a^{6}+\frac{1411}{736}a^{5}-\frac{33797}{6624}a^{4}+\frac{977}{828}a^{3}-\frac{1591}{1104}a^{2}-\frac{170}{207}a+\frac{70}{207}$, $\frac{7}{552}a^{16}-\frac{15}{368}a^{13}+\frac{451}{1104}a^{10}-\frac{3595}{1104}a^{7}+\frac{2665}{368}a^{4}-\frac{143}{276}a+2$, $\frac{41}{1104}a^{17}-\frac{7}{368}a^{15}-\frac{35}{276}a^{14}+\frac{7}{138}a^{12}+\frac{105}{92}a^{11}-\frac{107}{184}a^{9}-\frac{5375}{552}a^{8}+\frac{103}{23}a^{6}+\frac{22627}{1104}a^{5}-\frac{9313}{1104}a^{3}+\frac{827}{184}a^{2}-\frac{77}{23}$, $\frac{89}{828}a^{17}+\frac{5}{72}a^{16}+\frac{5}{368}a^{15}-\frac{559}{1656}a^{14}-\frac{17}{72}a^{13}-\frac{21}{368}a^{12}+\frac{2711}{828}a^{11}+\frac{155}{72}a^{10}+\frac{475}{1104}a^{9}-\frac{11321}{414}a^{8}-\frac{1315}{72}a^{7}-\frac{1463}{368}a^{6}+\frac{22343}{414}a^{5}+\frac{353}{9}a^{4}+\frac{1923}{184}a^{3}+\frac{30799}{1656}a^{2}+\frac{89}{18}a-\frac{341}{69}$, $\frac{91}{1104}a^{17}+\frac{197}{1656}a^{16}+\frac{3}{368}a^{15}-\frac{175}{552}a^{14}-\frac{1273}{3312}a^{13}-\frac{1}{1104}a^{12}+\frac{981}{368}a^{11}+\frac{12055}{3312}a^{10}+\frac{193}{1104}a^{9}-\frac{25127}{1104}a^{8}-\frac{101075}{3312}a^{7}-\frac{473}{368}a^{6}+\frac{30509}{552}a^{5}+\frac{205163}{3312}a^{4}-\frac{1019}{552}a^{3}-\frac{4605}{368}a^{2}+\frac{8369}{414}a+\frac{904}{69}$, $\frac{53}{828}a^{17}+\frac{7}{144}a^{16}-\frac{1}{368}a^{15}-\frac{367}{1656}a^{14}-\frac{25}{144}a^{13}-\frac{5}{368}a^{12}+\frac{410}{207}a^{11}+\frac{217}{144}a^{10}-\frac{49}{1104}a^{9}-\frac{6965}{414}a^{8}-\frac{1859}{144}a^{7}+\frac{81}{368}a^{6}+\frac{30229}{828}a^{5}+\frac{1043}{36}a^{4}+\frac{181}{46}a^{3}+\frac{14815}{1656}a^{2}+\frac{155}{18}a+\frac{59}{69}$ 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:  \( 47479719.21338162 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 47479719.21338162 \cdot 4}{2\cdot\sqrt{13741574276458659759521484375}}\cr\approx \mathstrut & 12.3634224200745 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^15 + 30*x^12 - 250*x^9 + 465*x^6 + 237*x^3 + 64)
 
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 - 3*x^15 + 30*x^12 - 250*x^9 + 465*x^6 + 237*x^3 + 64, 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 - 3*x^15 + 30*x^12 - 250*x^9 + 465*x^6 + 237*x^3 + 64);
 
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 - 3*x^15 + 30*x^12 - 250*x^9 + 465*x^6 + 237*x^3 + 64);
 
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

$C_6:S_3$ (as 18T12):

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 36
The 12 conjugacy class representatives for $C_6:S_3$
Character table for $C_6:S_3$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.6075.2, 3.1.243.1, 3.1.675.1, 3.1.6075.1, 6.0.553584375.2, 6.0.6834375.1, 6.0.22143375.1, 6.0.553584375.1, 9.1.6053445140625.1

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

Galois closure: data not computed
Degree 18 sibling: 18.2.4580524758819553253173828125.2
Minimal sibling: 18.2.4580524758819553253173828125.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{8}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ R R ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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
\(3\) Copy content Toggle raw display Deg $18$$18$$1$$37$
\(5\) Copy content Toggle raw display 5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 20 x^{7} + 10 x^{6} + 50 x^{2} + 100 x + 25$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$