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Properties

Label 18.18.1287206634...3125.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{21}\cdot 5^{12}\cdot 23^{13}$
Root discriminant $101.41$
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![63075, -19575, -589545, 345060, 1609968, -1310592, -1444974, 1574322, 318291, -693201, 67869, 118758, -25149, -8727, 2358, 273, -84, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 84*x^16 + 273*x^15 + 2358*x^14 - 8727*x^13 - 25149*x^12 + 118758*x^11 + 67869*x^10 - 693201*x^9 + 318291*x^8 + 1574322*x^7 - 1444974*x^6 - 1310592*x^5 + 1609968*x^4 + 345060*x^3 - 589545*x^2 - 19575*x + 63075)
 
gp: K = bnfinit(x^18 - 3*x^17 - 84*x^16 + 273*x^15 + 2358*x^14 - 8727*x^13 - 25149*x^12 + 118758*x^11 + 67869*x^10 - 693201*x^9 + 318291*x^8 + 1574322*x^7 - 1444974*x^6 - 1310592*x^5 + 1609968*x^4 + 345060*x^3 - 589545*x^2 - 19575*x + 63075, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 84 x^{16} + 273 x^{15} + 2358 x^{14} - 8727 x^{13} - 25149 x^{12} + 118758 x^{11} + 67869 x^{10} - 693201 x^{9} + 318291 x^{8} + 1574322 x^{7} - 1444974 x^{6} - 1310592 x^{5} + 1609968 x^{4} + 345060 x^{3} - 589545 x^{2} - 19575 x + 63075 \)

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:  \(1287206634035789519596942809814453125=3^{21}\cdot 5^{12}\cdot 23^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{9} - \frac{2}{5} a^{7} + \frac{1}{10} a^{6} - \frac{3}{10} a^{5} - \frac{1}{5} a^{4} - \frac{3}{10} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{50} a^{13} + \frac{1}{25} a^{12} + \frac{1}{50} a^{11} - \frac{1}{25} a^{10} - \frac{1}{25} a^{9} - \frac{1}{50} a^{8} + \frac{3}{50} a^{7} + \frac{12}{25} a^{6} - \frac{23}{50} a^{5} - \frac{23}{50} a^{4} - \frac{1}{2} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{1150} a^{14} + \frac{1}{115} a^{13} - \frac{3}{1150} a^{12} - \frac{27}{575} a^{11} + \frac{11}{575} a^{10} - \frac{67}{1150} a^{9} - \frac{3}{230} a^{8} + \frac{149}{575} a^{7} - \frac{551}{1150} a^{6} - \frac{477}{1150} a^{5} - \frac{89}{1150} a^{4} + \frac{10}{23} a^{3} + \frac{4}{23} a^{2} + \frac{2}{23} a + \frac{9}{23}$, $\frac{1}{1150} a^{15} - \frac{11}{1150} a^{13} + \frac{9}{230} a^{12} - \frac{18}{575} a^{11} - \frac{11}{1150} a^{10} + \frac{11}{1150} a^{9} + \frac{11}{1150} a^{8} - \frac{3}{23} a^{7} - \frac{232}{575} a^{6} - \frac{54}{115} a^{5} - \frac{18}{575} a^{4} + \frac{15}{46} a^{3} - \frac{52}{115} a^{2} + \frac{1}{46} a - \frac{19}{46}$, $\frac{1}{5750} a^{16} - \frac{1}{5750} a^{15} - \frac{1}{2875} a^{14} - \frac{3}{1150} a^{13} + \frac{26}{575} a^{12} + \frac{183}{5750} a^{11} + \frac{41}{2875} a^{10} + \frac{147}{2875} a^{9} + \frac{19}{1150} a^{8} + \frac{377}{1150} a^{7} + \frac{1358}{2875} a^{6} - \frac{431}{5750} a^{5} + \frac{1933}{5750} a^{4} - \frac{521}{1150} a^{3} + \frac{108}{575} a^{2} - \frac{7}{230} a + \frac{43}{230}$, $\frac{1}{313820503898426200188908949250} a^{17} - \frac{1104007819839329090527701}{31382050389842620018890894925} a^{16} + \frac{60285629086381606666594416}{156910251949213100094454474625} a^{15} + \frac{568253430565444163080871}{13644369734714182616909084750} a^{14} + \frac{45556904280632962451356258}{31382050389842620018890894925} a^{13} + \frac{5584061980135910881332343749}{156910251949213100094454474625} a^{12} + \frac{506949684622636657338919859}{12552820155937048007556357970} a^{11} + \frac{3446395050354312594238339448}{156910251949213100094454474625} a^{10} + \frac{9777922587657937232310274222}{156910251949213100094454474625} a^{9} - \frac{619540412358536177531717348}{31382050389842620018890894925} a^{8} - \frac{101064773481708222127144460329}{313820503898426200188908949250} a^{7} - \frac{13501484063828341390288534823}{62764100779685240037781789850} a^{6} - \frac{109922946579523026685707574803}{313820503898426200188908949250} a^{5} + \frac{30940539636129231569433894023}{313820503898426200188908949250} a^{4} + \frac{211865002728427565511148018}{6276410077968524003778178985} a^{3} - \frac{5041539819140663173052853617}{31382050389842620018890894925} a^{2} + \frac{840866217544802763451497243}{6276410077968524003778178985} a - \frac{157206080042139714091499103}{432855867446105103708839930}$

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:  \( 15141979627900 \) (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

\(\Q(\sqrt{69}) \), 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.2$x^{4} - 20$$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$$[\ ]_{4}$
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{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}$