↑

Properties

Label 18.0.38300347063...5792.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 19^{16}\cdot 37^{3}$
Root discriminant $50.01$
Ramified primes $2, 19, 37$
Class number $2280$ (GRH)
Class group $[2, 2, 570]$ (GRH)
Galois group $C_2\times C_2^2:C_9$ (as 18T26)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10525279, -14690014, 13279313, -8338866, 4885022, -2158328, 834376, -244982, 150518, -96972, 54321, -20338, 8164, -2334, 667, -124, 36, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 36*x^16 - 124*x^15 + 667*x^14 - 2334*x^13 + 8164*x^12 - 20338*x^11 + 54321*x^10 - 96972*x^9 + 150518*x^8 - 244982*x^7 + 834376*x^6 - 2158328*x^5 + 4885022*x^4 - 8338866*x^3 + 13279313*x^2 - 14690014*x + 10525279)
 
gp: K = bnfinit(x^18 - 2*x^17 + 36*x^16 - 124*x^15 + 667*x^14 - 2334*x^13 + 8164*x^12 - 20338*x^11 + 54321*x^10 - 96972*x^9 + 150518*x^8 - 244982*x^7 + 834376*x^6 - 2158328*x^5 + 4885022*x^4 - 8338866*x^3 + 13279313*x^2 - 14690014*x + 10525279, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 36 x^{16} - 124 x^{15} + 667 x^{14} - 2334 x^{13} + 8164 x^{12} - 20338 x^{11} + 54321 x^{10} - 96972 x^{9} + 150518 x^{8} - 244982 x^{7} + 834376 x^{6} - 2158328 x^{5} + 4885022 x^{4} - 8338866 x^{3} + 13279313 x^{2} - 14690014 x + 10525279 \)

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:  \(-3830034706318154794675914145792=-\,2^{18}\cdot 19^{16}\cdot 37^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.01$
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:  $2, 19, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{427965565405381003973147713234418149013512522259760603108649} a^{17} + \frac{23088252600276962406494145315410524806261523752312407615935}{427965565405381003973147713234418149013512522259760603108649} a^{16} + \frac{123361099195093731491608097583715764154574036774675483938409}{427965565405381003973147713234418149013512522259760603108649} a^{15} + \frac{138742465218227853418278752113608779784671454461836228498074}{427965565405381003973147713234418149013512522259760603108649} a^{14} + \frac{185286679536708998514572557861331270497497327945114292064176}{427965565405381003973147713234418149013512522259760603108649} a^{13} + \frac{25539178907294385362605481464882332239285253690595750148492}{427965565405381003973147713234418149013512522259760603108649} a^{12} - \frac{173440919964409882429153647339746531194253324347490012052507}{427965565405381003973147713234418149013512522259760603108649} a^{11} + \frac{15413662693260872082522625339221290784330686538439529944}{2834209042419741748166541147247802311347765048077884788799} a^{10} - \frac{119857623160877445139732872936721386025597625604959079276801}{427965565405381003973147713234418149013512522259760603108649} a^{9} + \frac{75485753915723953479155237709595202871532112862105822057068}{427965565405381003973147713234418149013512522259760603108649} a^{8} + \frac{94707808519032168101301461022607151697551036346726141470893}{427965565405381003973147713234418149013512522259760603108649} a^{7} - \frac{155145122178330151504908882378726352874358119479083093855605}{427965565405381003973147713234418149013512522259760603108649} a^{6} + \frac{23151384295532178886522119285206028557438112284088644439091}{427965565405381003973147713234418149013512522259760603108649} a^{5} + \frac{189791498748932586269788118236761242358359791494464054461676}{427965565405381003973147713234418149013512522259760603108649} a^{4} - \frac{205660814162934966825048356920499044101399853115336288955427}{427965565405381003973147713234418149013512522259760603108649} a^{3} + \frac{83449925860830679796563980500100231221411097844647829531883}{427965565405381003973147713234418149013512522259760603108649} a^{2} - \frac{206453826673285230041644276349407303102891433377101953688732}{427965565405381003973147713234418149013512522259760603108649} a - \frac{2839851385497415843533886334477564632809991364076152443650}{11566636902848135242517505763092382405770608709723259543477}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{570}$, which has order $2280$ (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:  \( -1 \) (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:  \( 22305.8950792 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:C_9$ (as 18T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 24 conjugacy class representatives for $C_2\times C_2^2:C_9$
Character table for $C_2\times C_2^2:C_9$ is not computed

Intermediate fields

3.3.361.1, 6.0.308600128.1, \(\Q(\zeta_{19})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $18$ $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ $18$ $18$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$