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Properties

Label 18.0.85493948743...2347.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 7^{12}$
Root discriminant $24.27$
Ramified primes $3, 7$
Class number $3$
Class group $[3]$
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![2512, -15720, 47184, -95718, 150321, -188391, 198099, -175122, 133758, -87574, 50088, -24600, 10548, -3822, 1194, -300, 63, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 63*x^16 - 300*x^15 + 1194*x^14 - 3822*x^13 + 10548*x^12 - 24600*x^11 + 50088*x^10 - 87574*x^9 + 133758*x^8 - 175122*x^7 + 198099*x^6 - 188391*x^5 + 150321*x^4 - 95718*x^3 + 47184*x^2 - 15720*x + 2512)
 
gp: K = bnfinit(x^18 - 9*x^17 + 63*x^16 - 300*x^15 + 1194*x^14 - 3822*x^13 + 10548*x^12 - 24600*x^11 + 50088*x^10 - 87574*x^9 + 133758*x^8 - 175122*x^7 + 198099*x^6 - 188391*x^5 + 150321*x^4 - 95718*x^3 + 47184*x^2 - 15720*x + 2512, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 63 x^{16} - 300 x^{15} + 1194 x^{14} - 3822 x^{13} + 10548 x^{12} - 24600 x^{11} + 50088 x^{10} - 87574 x^{9} + 133758 x^{8} - 175122 x^{7} + 198099 x^{6} - 188391 x^{5} + 150321 x^{4} - 95718 x^{3} + 47184 x^{2} - 15720 x + 2512 \)

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:  \(-8549394874383196572862347=-\,3^{31}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.27$
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:  $3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4}$, $\frac{1}{128} a^{14} - \frac{7}{128} a^{13} - \frac{3}{128} a^{12} + \frac{13}{128} a^{11} + \frac{13}{128} a^{10} - \frac{15}{128} a^{9} - \frac{13}{128} a^{8} + \frac{27}{128} a^{7} - \frac{17}{128} a^{6} + \frac{7}{128} a^{5} - \frac{9}{128} a^{4} - \frac{13}{128} a^{3} + \frac{1}{4} a^{2} - \frac{1}{8} a - \frac{5}{16}$, $\frac{1}{128} a^{15} - \frac{1}{32} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{11} - \frac{1}{32} a^{10} + \frac{5}{64} a^{9} - \frac{1}{32} a^{7} + \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{1}{32} a^{4} - \frac{59}{128} a^{3} + \frac{1}{8} a^{2} - \frac{3}{16} a - \frac{3}{16}$, $\frac{1}{256} a^{16} + \frac{3}{64} a^{13} - \frac{1}{64} a^{12} + \frac{1}{16} a^{11} - \frac{9}{128} a^{10} + \frac{1}{64} a^{9} + \frac{1}{32} a^{8} + \frac{3}{64} a^{7} + \frac{1}{64} a^{6} - \frac{1}{4} a^{5} + \frac{49}{256} a^{4} - \frac{9}{64} a^{3} - \frac{11}{32} a^{2} + \frac{5}{32} a + \frac{3}{8}$, $\frac{1}{789248} a^{17} + \frac{1533}{789248} a^{16} - \frac{89}{98656} a^{15} - \frac{41}{49328} a^{14} + \frac{11007}{197312} a^{13} + \frac{1137}{49328} a^{12} + \frac{12905}{394624} a^{11} - \frac{19721}{394624} a^{10} + \frac{5559}{49328} a^{9} - \frac{1925}{24664} a^{8} + \frac{45779}{197312} a^{7} + \frac{10963}{49328} a^{6} - \frac{90323}{789248} a^{5} + \frac{14629}{789248} a^{4} - \frac{36753}{98656} a^{3} + \frac{24523}{49328} a^{2} + \frac{5757}{98656} a - \frac{405}{3083}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{525}{197312} a^{17} + \frac{8925}{394624} a^{16} - \frac{30073}{197312} a^{15} + \frac{272595}{394624} a^{14} - \frac{1029537}{394624} a^{13} + \frac{3099083}{394624} a^{12} - \frac{7945821}{394624} a^{11} + \frac{17005989}{394624} a^{10} - \frac{31086853}{394624} a^{9} + \frac{47819073}{394624} a^{8} - \frac{61777659}{394624} a^{7} + \frac{65800489}{394624} a^{6} - \frac{55614333}{394624} a^{5} + \frac{17828517}{197312} a^{4} - \frac{13802845}{394624} a^{3} + \frac{138867}{49328} a^{2} + \frac{420201}{49328} a - \frac{151429}{49328} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 980950.022268 \)
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

\(\Q(\sqrt{-3}) \), 3.1.1323.1 x3, 3.3.3969.1, 6.0.5250987.1, 6.0.964467.3 x2, 6.0.47258883.2, 9.3.1688134559643.5 x3

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

Sibling fields

Degree 6 sibling: 6.0.964467.3
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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

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
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$