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Properties

Label 18.0.21458931758...9363.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{12}\cdot 31^{12}$
Root discriminant $62.55$
Ramified primes $3, 7, 31$
Class number $729$ (GRH)
Class group $[3, 3, 3, 3, 9]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22707, 30537, -14238, -37557, 27066, 816, -15371, 5367, 12519, -16353, 10920, -4929, 1582, -525, 243, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 243*x^14 - 525*x^13 + 1582*x^12 - 4929*x^11 + 10920*x^10 - 16353*x^9 + 12519*x^8 + 5367*x^7 - 15371*x^6 + 816*x^5 + 27066*x^4 - 37557*x^3 - 14238*x^2 + 30537*x + 22707)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 243*x^14 - 525*x^13 + 1582*x^12 - 4929*x^11 + 10920*x^10 - 16353*x^9 + 12519*x^8 + 5367*x^7 - 15371*x^6 + 816*x^5 + 27066*x^4 - 37557*x^3 - 14238*x^2 + 30537*x + 22707, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 243 x^{14} - 525 x^{13} + 1582 x^{12} - 4929 x^{11} + 10920 x^{10} - 16353 x^{9} + 12519 x^{8} + 5367 x^{7} - 15371 x^{6} + 816 x^{5} + 27066 x^{4} - 37557 x^{3} - 14238 x^{2} + 30537 x + 22707 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-214589317581007476486780131079363=-\,3^{9}\cdot 7^{12}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.55$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{87} a^{10} - \frac{5}{87} a^{9} + \frac{4}{29} a^{8} + \frac{11}{87} a^{7} + \frac{3}{29} a^{6} - \frac{43}{87} a^{5} - \frac{4}{87} a^{4} + \frac{1}{87} a^{3} - \frac{14}{29} a^{2} - \frac{9}{29} a$, $\frac{1}{87} a^{11} - \frac{13}{87} a^{9} + \frac{13}{87} a^{8} + \frac{2}{29} a^{7} + \frac{2}{87} a^{6} - \frac{16}{87} a^{5} - \frac{19}{87} a^{4} - \frac{8}{87} a^{3} - \frac{5}{87} a^{2} + \frac{13}{29} a$, $\frac{1}{261} a^{12} - \frac{1}{261} a^{10} - \frac{2}{29} a^{9} - \frac{8}{87} a^{8} + \frac{2}{29} a^{7} + \frac{5}{261} a^{6} + \frac{5}{29} a^{5} - \frac{56}{261} a^{4} + \frac{4}{29} a^{3} + \frac{19}{87} a^{2} - \frac{7}{29} a$, $\frac{1}{4959} a^{13} + \frac{1}{1653} a^{12} + \frac{11}{4959} a^{11} - \frac{3}{551} a^{10} + \frac{74}{551} a^{9} - \frac{193}{1653} a^{8} + \frac{587}{4959} a^{7} + \frac{13}{551} a^{6} + \frac{5}{171} a^{5} + \frac{4}{1653} a^{4} + \frac{427}{1653} a^{3} + \frac{71}{1653} a^{2} - \frac{125}{551} a + \frac{7}{19}$, $\frac{1}{143811} a^{14} - \frac{7}{143811} a^{13} - \frac{4}{2523} a^{12} - \frac{194}{143811} a^{11} + \frac{575}{143811} a^{10} + \frac{89}{2523} a^{9} - \frac{14371}{143811} a^{8} - \frac{19433}{143811} a^{7} + \frac{5276}{47937} a^{6} + \frac{60863}{143811} a^{5} - \frac{10220}{143811} a^{4} + \frac{370}{2523} a^{3} + \frac{4496}{15979} a^{2} + \frac{170}{551} a - \frac{7}{19}$, $\frac{1}{1006677} a^{15} + \frac{1}{335559} a^{14} - \frac{8}{1006677} a^{13} + \frac{1151}{1006677} a^{12} - \frac{4787}{1006677} a^{11} + \frac{34}{143811} a^{10} - \frac{117631}{1006677} a^{9} - \frac{5744}{111853} a^{8} - \frac{94228}{1006677} a^{7} + \frac{109813}{1006677} a^{6} + \frac{69533}{143811} a^{5} + \frac{325151}{1006677} a^{4} + \frac{162436}{335559} a^{3} - \frac{2020}{111853} a^{2} + \frac{71}{203} a - \frac{2}{7}$, $\frac{1}{29193633} a^{16} - \frac{8}{29193633} a^{15} + \frac{50}{29193633} a^{14} - \frac{10}{1390173} a^{13} + \frac{48079}{29193633} a^{12} + \frac{49451}{29193633} a^{11} - \frac{42143}{29193633} a^{10} - \frac{1532947}{29193633} a^{9} + \frac{190381}{1536507} a^{8} + \frac{1446548}{9731211} a^{7} + \frac{3624878}{29193633} a^{6} - \frac{7349096}{29193633} a^{5} - \frac{10142462}{29193633} a^{4} + \frac{1211621}{9731211} a^{3} - \frac{92021}{1390173} a^{2} + \frac{21782}{111853} a + \frac{1314}{3857}$, $\frac{1}{3824365923} a^{17} + \frac{1}{67094139} a^{16} + \frac{1705}{3824365923} a^{15} + \frac{8}{1274788641} a^{14} + \frac{360505}{3824365923} a^{13} + \frac{4656718}{3824365923} a^{12} + \frac{10406222}{3824365923} a^{11} - \frac{3960028}{3824365923} a^{10} + \frac{7148375}{3824365923} a^{9} - \frac{40650049}{424929547} a^{8} + \frac{392342162}{3824365923} a^{7} + \frac{490745930}{3824365923} a^{6} + \frac{635671244}{1274788641} a^{5} - \frac{599915732}{3824365923} a^{4} - \frac{15375212}{1274788641} a^{3} + \frac{160243978}{424929547} a^{2} - \frac{4169657}{14652743} a - \frac{149084}{505267}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}$, which has order $729$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{152578}{3824365923} a^{17} - \frac{1296913}{3824365923} a^{16} + \frac{96548}{67094139} a^{15} - \frac{15336010}{3824365923} a^{14} + \frac{11914318}{1274788641} a^{13} - \frac{78412438}{3824365923} a^{12} + \frac{238098010}{3824365923} a^{11} - \frac{711435692}{3824365923} a^{10} + \frac{2581730}{6279747} a^{9} - \frac{2493119564}{3824365923} a^{8} + \frac{747754706}{1274788641} a^{7} - \frac{7592530}{131874687} a^{6} - \frac{1105719008}{3824365923} a^{5} + \frac{108795455}{1274788641} a^{4} + \frac{470710210}{424929547} a^{3} - \frac{663972126}{424929547} a^{2} - \frac{1467805}{14652743} a + \frac{58046}{72181} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8633591.79685 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.141267.1 x3, 3.3.47089.1, 6.0.59869095867.1, 6.0.59869095867.3, 6.0.1271403.1 x2, 9.3.2819175855281163.5 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: 6.0.1271403.1
Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
$31$31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$