Properties

Label 18.0.139...207.1
Degree $18$
Signature $[0, 9]$
Discriminant $-1.395\times 10^{19}$
Root discriminant \(11.58\)
Ramified primes $11,23$
Class number $1$
Class group trivial
Galois group $C_3^3:S_4$ (as 18T217)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{18} - x^{17} + 4 x^{15} - 5 x^{14} + x^{13} + 6 x^{12} - 9 x^{11} + 7 x^{10} + x^{9} - 12 x^{8} + 11 x^{7} - 2 x^{6} - 5 x^{5} + 6 x^{4} - x^{3} - x^{2} - x + 1 \) 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:   \(-13950087665617805207\) \(\medspace = -\,11^{4}\cdot 23^{11}\) 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:  \(11.58\)
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:  $11^{1/2}23^{3/4}\approx 34.833107461122644$
Ramified primes:   \(11\), \(23\) 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:  \(\Q(\sqrt{-23}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{16867}a^{17}-\frac{6775}{16867}a^{16}-\frac{1257}{16867}a^{15}-\frac{2913}{16867}a^{14}-\frac{1733}{16867}a^{13}-\frac{89}{16867}a^{12}-\frac{4320}{16867}a^{11}-\frac{574}{16867}a^{10}-\frac{7994}{16867}a^{9}+\frac{8287}{16867}a^{8}-\frac{2774}{16867}a^{7}+\frac{1249}{16867}a^{6}+\frac{6506}{16867}a^{5}+\frac{1822}{16867}a^{4}+\frac{4422}{16867}a^{3}+\frac{1163}{16867}a^{2}-\frac{1274}{16867}a-\frac{5829}{16867}$ 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:   $\frac{17410}{16867}a^{17}-\frac{1819}{16867}a^{16}+\frac{8996}{16867}a^{15}+\frac{54340}{16867}a^{14}-\frac{13334}{16867}a^{13}+\frac{19141}{16867}a^{12}+\frac{49354}{16867}a^{11}-\frac{24943}{16867}a^{10}+\frac{61545}{16867}a^{9}+\frac{13219}{16867}a^{8}-\frac{55720}{16867}a^{7}-\frac{13340}{16867}a^{6}-\frac{9312}{16867}a^{5}-\frac{5807}{16867}a^{4}-\frac{27702}{16867}a^{3}+\frac{41164}{16867}a^{2}-\frac{235}{16867}a-\frac{11018}{16867}$, $\frac{5790}{16867}a^{17}+\frac{5392}{16867}a^{16}-\frac{8353}{16867}a^{15}+\frac{34464}{16867}a^{14}+\frac{1795}{16867}a^{13}-\frac{26167}{16867}a^{12}+\frac{68429}{16867}a^{11}-\frac{34395}{16867}a^{10}-\frac{2212}{16867}a^{9}+\frac{79450}{16867}a^{8}-\frac{105278}{16867}a^{7}+\frac{29501}{16867}a^{6}+\frac{22596}{16867}a^{5}-\frac{76830}{16867}a^{4}+\frac{33008}{16867}a^{3}-\frac{13030}{16867}a^{2}+\frac{11286}{16867}a+\frac{957}{16867}$, $\frac{15616}{16867}a^{17}-\frac{8576}{16867}a^{16}+\frac{3876}{16867}a^{15}+\frac{51492}{16867}a^{14}-\frac{41594}{16867}a^{13}+\frac{10137}{16867}a^{12}+\frac{57481}{16867}a^{11}-\frac{74675}{16867}a^{10}+\frac{65831}{16867}a^{9}+\frac{6168}{16867}a^{8}-\frac{122397}{16867}a^{7}+\frac{56733}{16867}a^{6}-\frac{9112}{16867}a^{5}-\frac{52878}{16867}a^{4}+\frac{34188}{16867}a^{3}+\frac{12516}{16867}a^{2}-\frac{8591}{16867}a+\frac{5535}{16867}$, $\frac{9857}{16867}a^{17}-\frac{4722}{16867}a^{16}+\frac{6996}{16867}a^{15}+\frac{27927}{16867}a^{14}-\frac{12777}{16867}a^{13}+\frac{16678}{16867}a^{12}+\frac{23802}{16867}a^{11}-\frac{7473}{16867}a^{10}+\frac{39500}{16867}a^{9}-\frac{1922}{16867}a^{8}-\frac{18778}{16867}a^{7}-\frac{1517}{16867}a^{6}+\frac{18175}{16867}a^{5}-\frac{3901}{16867}a^{4}-\frac{30408}{16867}a^{3}+\frac{27865}{16867}a^{2}-\frac{8770}{16867}a-\frac{7451}{16867}$, $\frac{2247}{16867}a^{17}+\frac{7476}{16867}a^{16}-\frac{7690}{16867}a^{15}+\frac{15752}{16867}a^{14}+\frac{19093}{16867}a^{13}-\frac{31313}{16867}a^{12}+\frac{42086}{16867}a^{11}+\frac{8981}{16867}a^{10}-\frac{32897}{16867}a^{9}+\frac{84056}{16867}a^{8}-\frac{42989}{16867}a^{7}-\frac{27153}{16867}a^{6}+\frac{62761}{16867}a^{5}-\frac{55248}{16867}a^{4}+\frac{18438}{16867}a^{3}+\frac{32610}{16867}a^{2}-\frac{12155}{16867}a-\frac{8971}{16867}$, $\frac{9221}{16867}a^{17}+\frac{3093}{16867}a^{16}-\frac{3168}{16867}a^{15}+\frac{42092}{16867}a^{14}-\frac{6944}{16867}a^{13}-\frac{11053}{16867}a^{12}+\frac{72602}{16867}a^{11}-\frac{47217}{16867}a^{10}+\frac{29850}{16867}a^{9}+\frac{74385}{16867}a^{8}-\frac{109884}{16867}a^{7}+\frac{47469}{16867}a^{6}+\frac{12774}{16867}a^{5}-\frac{66338}{16867}a^{4}+\frac{58324}{16867}a^{3}+\frac{13478}{16867}a^{2}+\frac{8745}{16867}a-\frac{10947}{16867}$, $\frac{9801}{16867}a^{17}+\frac{3604}{16867}a^{16}-\frac{6947}{16867}a^{15}+\frac{39252}{16867}a^{14}-\frac{64}{16867}a^{13}-\frac{28939}{16867}a^{12}+\frac{63318}{16867}a^{11}-\frac{25930}{16867}a^{10}-\frac{1979}{16867}a^{9}+\frac{73750}{16867}a^{8}-\frac{99572}{16867}a^{7}-\frac{3993}{16867}a^{6}+\frac{58647}{16867}a^{5}-\frac{55332}{16867}a^{4}+\frac{25566}{16867}a^{3}+\frac{30205}{16867}a^{2}-\frac{21761}{16867}a-\frac{1500}{16867}$, $\frac{17410}{16867}a^{17}-\frac{1819}{16867}a^{16}+\frac{8996}{16867}a^{15}+\frac{54340}{16867}a^{14}-\frac{13334}{16867}a^{13}+\frac{19141}{16867}a^{12}+\frac{49354}{16867}a^{11}-\frac{24943}{16867}a^{10}+\frac{61545}{16867}a^{9}+\frac{13219}{16867}a^{8}-\frac{55720}{16867}a^{7}-\frac{13340}{16867}a^{6}-\frac{9312}{16867}a^{5}-\frac{5807}{16867}a^{4}-\frac{27702}{16867}a^{3}+\frac{41164}{16867}a^{2}+\frac{16632}{16867}a-\frac{11018}{16867}$ 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:  \( 91.5286994248 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 91.5286994248 \cdot 1}{2\cdot\sqrt{13950087665617805207}}\cr\approx \mathstrut & 0.187006904927 \end{aligned}\]

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

$C_3^3:S_4$ (as 18T217):

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 648
The 14 conjugacy class representatives for $C_3^3:S_4$
Character table for $C_3^3:S_4$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.33860761.1

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 9 sibling: data not computed
Degree 12 siblings: 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: 9.1.33860761.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.9.0.1}{9} }^{2}$ ${\href{/padicField/3.9.0.1}{9} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ R ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ R ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.9.0.1}{9} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ ${\href{/padicField/47.9.0.1}{9} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(11\) Copy content Toggle raw display 11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.2$x^{4} - 77 x^{2} + 242$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 77 x^{2} + 242$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.6.0.1$x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
\(23\) Copy content Toggle raw display 23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$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}$
23.4.3.1$x^{4} + 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$