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Properties

Label 18.2.11777310696...4672.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{50}\cdot 3^{21}$
Root discriminant $24.71$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $PSU(3,2)$ (as 18T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, 162, -351, 0, 1782, -4860, 7614, -8208, 6498, -3780, 1674, -576, 198, -108, 78, -48, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 48*x^15 + 78*x^14 - 108*x^13 + 198*x^12 - 576*x^11 + 1674*x^10 - 3780*x^9 + 6498*x^8 - 8208*x^7 + 7614*x^6 - 4860*x^5 + 1782*x^4 - 351*x^2 + 162*x - 27)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 - 48*x^15 + 78*x^14 - 108*x^13 + 198*x^12 - 576*x^11 + 1674*x^10 - 3780*x^9 + 6498*x^8 - 8208*x^7 + 7614*x^6 - 4860*x^5 + 1782*x^4 - 351*x^2 + 162*x - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} - 48 x^{15} + 78 x^{14} - 108 x^{13} + 198 x^{12} - 576 x^{11} + 1674 x^{10} - 3780 x^{9} + 6498 x^{8} - 8208 x^{7} + 7614 x^{6} - 4860 x^{5} + 1782 x^{4} - 351 x^{2} + 162 x - 27 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11777310696798643575324672=2^{50}\cdot 3^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.71$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{18} a^{11} + \frac{1}{3} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{36} a^{12} - \frac{1}{12} a^{8} + \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{72} a^{13} - \frac{1}{72} a^{12} - \frac{1}{24} a^{9} + \frac{1}{24} a^{8} - \frac{3}{8} a^{5} + \frac{3}{8} a^{4} + \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{144} a^{14} - \frac{1}{144} a^{12} + \frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{48} a^{6} - \frac{5}{16} a^{4} + \frac{5}{16} a^{2} - \frac{5}{16}$, $\frac{1}{144} a^{15} - \frac{1}{144} a^{13} + \frac{1}{144} a^{11} - \frac{1}{16} a^{9} - \frac{1}{48} a^{7} + \frac{17}{48} a^{5} - \frac{3}{16} a^{3} - \frac{5}{16} a$, $\frac{1}{432} a^{16} - \frac{1}{18} a^{10} - \frac{1}{12} a^{8} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2} + \frac{1}{16}$, $\frac{1}{47994336} a^{17} + \frac{30665}{47994336} a^{16} + \frac{349}{205104} a^{15} + \frac{6119}{3999528} a^{14} - \frac{48431}{7999056} a^{13} + \frac{4295}{3999528} a^{12} - \frac{60533}{7999056} a^{11} - \frac{274501}{3999528} a^{10} + \frac{42769}{2666352} a^{9} - \frac{8309}{166647} a^{8} + \frac{438125}{2666352} a^{7} - \frac{11937}{444392} a^{6} + \frac{17677}{68368} a^{5} + \frac{115019}{1333176} a^{4} + \frac{405663}{888784} a^{3} + \frac{29103}{444392} a^{2} - \frac{439477}{1777568} a - \frac{678603}{1777568}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 1938173.05938 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$PSU(3,2)$ (as 18T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 6 conjugacy class representatives for $PSU(3,2)$
Character table for $PSU(3,2)$

Intermediate fields

\(\Q(\sqrt{3}) \), 9.1.990677827584.1

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

Sibling fields

Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: 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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.8.24.12$x^{8} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 30$$8$$1$$24$$Q_8$$[2, 3, 4]$
2.8.24.12$x^{8} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 30$$8$$1$$24$$Q_8$$[2, 3, 4]$
3Data not computed