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Properties

Label 18.18.9327584304...3125.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{20}\cdot 5^{13}\cdot 23^{12}$
Root discriminant $87.65$
Ramified primes $3, 5, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![2095, 48300, 224865, 180750, -588846, -768105, 512415, 779346, -256524, -321677, 91992, 57978, -16878, -4947, 1503, 198, -63, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 63*x^16 + 198*x^15 + 1503*x^14 - 4947*x^13 - 16878*x^12 + 57978*x^11 + 91992*x^10 - 321677*x^9 - 256524*x^8 + 779346*x^7 + 512415*x^6 - 768105*x^5 - 588846*x^4 + 180750*x^3 + 224865*x^2 + 48300*x + 2095)
 
gp: K = bnfinit(x^18 - 3*x^17 - 63*x^16 + 198*x^15 + 1503*x^14 - 4947*x^13 - 16878*x^12 + 57978*x^11 + 91992*x^10 - 321677*x^9 - 256524*x^8 + 779346*x^7 + 512415*x^6 - 768105*x^5 - 588846*x^4 + 180750*x^3 + 224865*x^2 + 48300*x + 2095, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 63 x^{16} + 198 x^{15} + 1503 x^{14} - 4947 x^{13} - 16878 x^{12} + 57978 x^{11} + 91992 x^{10} - 321677 x^{9} - 256524 x^{8} + 779346 x^{7} + 512415 x^{6} - 768105 x^{5} - 588846 x^{4} + 180750 x^{3} + 224865 x^{2} + 48300 x + 2095 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(93275843046071704318619044189453125=3^{20}\cdot 5^{13}\cdot 23^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.65$
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:  $3, 5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{11} + \frac{1}{12} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{12}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{11} + \frac{1}{24} a^{10} - \frac{1}{24} a^{9} + \frac{3}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{7}{24} a^{4} - \frac{1}{12} a^{2} - \frac{7}{24} a + \frac{7}{24}$, $\frac{1}{240} a^{14} + \frac{1}{240} a^{13} - \frac{1}{48} a^{12} - \frac{1}{60} a^{11} - \frac{1}{6} a^{10} + \frac{3}{20} a^{9} - \frac{3}{16} a^{8} + \frac{11}{40} a^{7} + \frac{3}{8} a^{6} - \frac{49}{240} a^{5} + \frac{29}{240} a^{4} + \frac{11}{24} a^{3} - \frac{13}{48} a^{2} + \frac{1}{6} a + \frac{5}{16}$, $\frac{1}{480} a^{15} - \frac{1}{80} a^{13} + \frac{1}{480} a^{12} - \frac{3}{40} a^{11} - \frac{1}{120} a^{10} - \frac{1}{480} a^{9} - \frac{43}{160} a^{8} - \frac{9}{20} a^{7} + \frac{101}{480} a^{6} + \frac{13}{80} a^{5} - \frac{53}{160} a^{4} + \frac{13}{96} a^{3} - \frac{9}{32} a^{2} + \frac{23}{96} a - \frac{31}{96}$, $\frac{1}{960} a^{16} - \frac{1}{960} a^{15} - \frac{1}{480} a^{14} + \frac{11}{960} a^{13} + \frac{23}{960} a^{12} - \frac{1}{15} a^{11} - \frac{157}{960} a^{10} + \frac{1}{10} a^{9} + \frac{71}{320} a^{8} - \frac{139}{960} a^{7} + \frac{97}{960} a^{6} + \frac{287}{960} a^{5} + \frac{5}{48} a^{4} + \frac{5}{12} a^{3} + \frac{31}{96} a^{2} + \frac{37}{96} a + \frac{25}{64}$, $\frac{1}{180190167226332848188800} a^{17} - \frac{37140830923086289321}{90095083613166424094400} a^{16} - \frac{10592565220877959743}{12012677815088856545920} a^{15} + \frac{64688503338493696973}{180190167226332848188800} a^{14} - \frac{302063990988532392601}{45047541806583212047200} a^{13} + \frac{6600419746508502641689}{180190167226332848188800} a^{12} - \frac{1261584017124347192509}{180190167226332848188800} a^{11} + \frac{22932261134187294524909}{180190167226332848188800} a^{10} - \frac{4490532767145002548473}{60063389075444282729600} a^{9} + \frac{3706990865538830007209}{11261885451645803011800} a^{8} + \frac{241755301561277659439}{1801901672263328481888} a^{7} - \frac{11769098284116867403779}{30031694537722141364800} a^{6} + \frac{35052348783180611854261}{180190167226332848188800} a^{5} - \frac{7188866137754603807381}{45047541806583212047200} a^{4} - \frac{3540581336523552503881}{18019016722633284818880} a^{3} + \frac{3863108911753206526547}{9009508361316642409440} a^{2} + \frac{4534127941555729196801}{36038033445266569637760} a + \frac{4325315272831498068887}{12012677815088856545920}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2264806866520 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 9.9.136583925149390625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.8.6.1$x^{8} + 299 x^{4} + 25921$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
23.8.6.1$x^{8} + 299 x^{4} + 25921$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$