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Properties

Label 18.0.650...928.1
Degree $18$
Signature $[0, 9]$
Discriminant $-6.510\times 10^{27}$
Root discriminant \(35.09\)
Ramified primes $2,3,7,11$
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 - 25*x^15 + 252*x^12 - 1245*x^9 + 2786*x^6 - 1715*x^3 + 343)
 
gp: K = bnfinit(y^18 - 25*y^15 + 252*y^12 - 1245*y^9 + 2786*y^6 - 1715*y^3 + 343, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 25*x^15 + 252*x^12 - 1245*x^9 + 2786*x^6 - 1715*x^3 + 343);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 25*x^15 + 252*x^12 - 1245*x^9 + 2786*x^6 - 1715*x^3 + 343)
 

\( x^{18} - 25x^{15} + 252x^{12} - 1245x^{9} + 2786x^{6} - 1715x^{3} + 343 \) 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:   \(-6509957846719213507476860928\) \(\medspace = -\,2^{12}\cdot 3^{27}\cdot 7^{6}\cdot 11^{6}\) 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:  \(35.09\)
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:  $2^{2/3}3^{31/18}7^{2/3}11^{1/2}\approx 127.78813364145171$
Ramified primes:   \(2\), \(3\), \(7\), \(11\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{3}{7}a^{9}+\frac{1}{7}a^{3}$, $\frac{1}{7}a^{13}+\frac{3}{7}a^{10}+\frac{1}{7}a^{4}$, $\frac{1}{7}a^{14}+\frac{3}{7}a^{11}+\frac{1}{7}a^{5}$, $\frac{1}{638617}a^{15}-\frac{35501}{638617}a^{12}+\frac{11572}{91231}a^{9}+\frac{77351}{638617}a^{6}+\frac{5137}{91231}a^{3}+\frac{5583}{13033}$, $\frac{1}{638617}a^{16}-\frac{35501}{638617}a^{13}+\frac{11572}{91231}a^{10}+\frac{77351}{638617}a^{7}+\frac{5137}{91231}a^{4}+\frac{5583}{13033}a$, $\frac{1}{638617}a^{17}-\frac{35501}{638617}a^{14}+\frac{11572}{91231}a^{11}+\frac{77351}{638617}a^{8}+\frac{5137}{91231}a^{5}+\frac{5583}{13033}a^{2}$ 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

$C_{3}$, 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:   \( -\frac{90}{13033} a^{15} + \frac{2005}{13033} a^{12} - \frac{17946}{13033} a^{9} + \frac{76230}{13033} a^{6} - \frac{134456}{13033} a^{3} + \frac{50439}{13033} \)  (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:   $\frac{103}{638617}a^{15}-\frac{7363}{638617}a^{12}+\frac{18946}{91231}a^{9}-\frac{973485}{638617}a^{6}+\frac{411814}{91231}a^{3}-\frac{24469}{13033}$, $\frac{6220}{638617}a^{15}-\frac{128431}{638617}a^{12}+\frac{152977}{91231}a^{9}-\frac{4225700}{638617}a^{6}+\frac{985732}{91231}a^{3}-\frac{58817}{13033}$, $\frac{90}{13033}a^{16}-\frac{838}{638617}a^{15}-\frac{2005}{13033}a^{13}+\frac{8532}{638617}a^{12}+\frac{17946}{13033}a^{10}-\frac{112}{13033}a^{9}-\frac{76230}{13033}a^{7}-\frac{319821}{638617}a^{6}+\frac{134456}{13033}a^{4}+\frac{204612}{91231}a^{3}-\frac{37406}{13033}a-\frac{12740}{13033}$, $\frac{90}{13033}a^{16}+\frac{838}{638617}a^{15}-\frac{2005}{13033}a^{13}-\frac{8532}{638617}a^{12}+\frac{17946}{13033}a^{10}+\frac{112}{13033}a^{9}-\frac{76230}{13033}a^{7}+\frac{319821}{638617}a^{6}+\frac{134456}{13033}a^{4}-\frac{204612}{91231}a^{3}-\frac{50439}{13033}a+\frac{12740}{13033}$, $\frac{8093}{638617}a^{17}-\frac{75}{13033}a^{16}-\frac{90}{13033}a^{15}-\frac{205636}{638617}a^{14}+\frac{13868}{91231}a^{13}+\frac{2005}{13033}a^{12}+\frac{296817}{91231}a^{11}-\frac{143784}{91231}a^{10}-\frac{17946}{13033}a^{9}-\frac{10060889}{638617}a^{8}+\frac{102624}{13033}a^{7}+\frac{76230}{13033}a^{6}+\frac{2943929}{91231}a^{5}-\frac{1557618}{91231}a^{4}-\frac{134456}{13033}a^{3}-\frac{119489}{13033}a^{2}+\frac{87648}{13033}a+\frac{24373}{13033}$, $\frac{95}{638617}a^{17}-\frac{35}{13033}a^{16}-\frac{90}{13033}a^{15}+\frac{2952}{638617}a^{14}+\frac{4734}{91231}a^{13}+\frac{2005}{13033}a^{12}-\frac{8465}{91231}a^{11}-\frac{35820}{91231}a^{10}-\frac{17946}{13033}a^{9}+\frac{323558}{638617}a^{8}+\frac{16612}{13033}a^{7}+\frac{76230}{13033}a^{6}-\frac{33305}{91231}a^{5}-\frac{77845}{91231}a^{4}-\frac{134456}{13033}a^{3}-\frac{43067}{13033}a^{2}-\frac{21656}{13033}a+\frac{24373}{13033}$, $\frac{32063}{638617}a^{17}+\frac{19702}{638617}a^{16}+\frac{19006}{638617}a^{15}-\frac{800455}{638617}a^{14}-\frac{520011}{638617}a^{13}-\frac{534916}{638617}a^{12}+\frac{161490}{13033}a^{11}+\frac{761189}{91231}a^{10}+\frac{116229}{13033}a^{9}-\frac{37323101}{638617}a^{8}-\frac{25954057}{638617}a^{7}-\frac{28700993}{638617}a^{6}+\frac{10449043}{91231}a^{5}+\frac{1079148}{13033}a^{4}+\frac{8657531}{91231}a^{3}-\frac{326351}{13033}a^{2}-\frac{288980}{13033}a-\frac{369112}{13033}$, $\frac{41777}{638617}a^{17}+\frac{9238}{638617}a^{16}-\frac{19006}{638617}a^{15}-\frac{1077682}{638617}a^{14}-\frac{165255}{638617}a^{13}+\frac{534916}{638617}a^{12}+\frac{224905}{13033}a^{11}+\frac{148833}{91231}a^{10}-\frac{116229}{13033}a^{9}-\frac{54191738}{638617}a^{8}-\frac{1959736}{638617}a^{7}+\frac{28700993}{638617}a^{6}+\frac{16337420}{91231}a^{5}-\frac{85295}{13033}a^{4}-\frac{8657531}{91231}a^{3}-\frac{818656}{13033}a^{2}+\frac{160569}{13033}a+\frac{369112}{13033}$ 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:  \( 2374820.8449431765 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 2374820.8449431765 \cdot 3}{6\cdot\sqrt{6509957846719213507476860928}}\cr\approx \mathstrut & 0.224610773592508 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 25*x^15 + 252*x^12 - 1245*x^9 + 2786*x^6 - 1715*x^3 + 343)
 
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 - 25*x^15 + 252*x^12 - 1245*x^9 + 2786*x^6 - 1715*x^3 + 343, 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 - 25*x^15 + 252*x^12 - 1245*x^9 + 2786*x^6 - 1715*x^3 + 343);
 
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 - 25*x^15 + 252*x^12 - 1245*x^9 + 2786*x^6 - 1715*x^3 + 343);
 
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

\(\Q(\sqrt{-3}) \), 3.1.44.1, 6.0.964467.2, 6.0.52272.1

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 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: 12.0.421862402155751149824.3

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 R ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.18.12.1$x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$$3$$6$$12$$S_3 \times C_3$$[\ ]_{3}^{6}$
\(3\) Copy content Toggle raw display Deg $18$$6$$3$$27$
\(7\) Copy content Toggle raw display 7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.4.1$x^{6} + 14 x^{3} - 245$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.0.1$x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
\(11\) 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}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$