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Properties

Label 18.2.26906369488...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{26}\cdot 3^{18}\cdot 5^{16}\cdot 7^{14}$
Root discriminant $155.08$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{18}$ (as 18T982)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1059046848, 377540976, 565097442, 209720586, -111814770, -54019896, -31691898, -1915326, 2502192, 1360730, 593271, 97293, 23611, -1092, 180, -198, 21, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 21*x^16 - 198*x^15 + 180*x^14 - 1092*x^13 + 23611*x^12 + 97293*x^11 + 593271*x^10 + 1360730*x^9 + 2502192*x^8 - 1915326*x^7 - 31691898*x^6 - 54019896*x^5 - 111814770*x^4 + 209720586*x^3 + 565097442*x^2 + 377540976*x - 1059046848)
 
gp: K = bnfinit(x^18 - 3*x^17 + 21*x^16 - 198*x^15 + 180*x^14 - 1092*x^13 + 23611*x^12 + 97293*x^11 + 593271*x^10 + 1360730*x^9 + 2502192*x^8 - 1915326*x^7 - 31691898*x^6 - 54019896*x^5 - 111814770*x^4 + 209720586*x^3 + 565097442*x^2 + 377540976*x - 1059046848, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 21 x^{16} - 198 x^{15} + 180 x^{14} - 1092 x^{13} + 23611 x^{12} + 97293 x^{11} + 593271 x^{10} + 1360730 x^{9} + 2502192 x^{8} - 1915326 x^{7} - 31691898 x^{6} - 54019896 x^{5} - 111814770 x^{4} + 209720586 x^{3} + 565097442 x^{2} + 377540976 x - 1059046848 \)

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:  \(2690636948830640160368640000000000000000=2^{26}\cdot 3^{18}\cdot 5^{16}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.08$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10}$, $\frac{1}{28} a^{14} + \frac{5}{28} a^{13} - \frac{5}{28} a^{12} + \frac{3}{28} a^{11} + \frac{13}{28} a^{10} - \frac{5}{28} a^{9} + \frac{3}{14} a^{8} - \frac{5}{14} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{14} a^{4} - \frac{1}{14} a^{3} - \frac{5}{14} a^{2} - \frac{1}{14} a + \frac{1}{7}$, $\frac{1}{28} a^{15} - \frac{1}{14} a^{13} - \frac{1}{14} a^{11} - \frac{1}{2} a^{10} + \frac{3}{28} a^{9} - \frac{3}{7} a^{8} - \frac{5}{14} a^{7} - \frac{1}{7} a^{6} - \frac{1}{2} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{2} - \frac{1}{2} a + \frac{2}{7}$, $\frac{1}{392} a^{16} - \frac{3}{196} a^{15} - \frac{1}{56} a^{14} - \frac{97}{392} a^{13} - \frac{89}{392} a^{12} + \frac{11}{392} a^{11} - \frac{31}{196} a^{10} - \frac{145}{392} a^{9} - \frac{41}{98} a^{8} + \frac{53}{196} a^{7} + \frac{15}{196} a^{6} - \frac{10}{49} a^{5} + \frac{81}{196} a^{4} + \frac{57}{196} a^{3} - \frac{1}{7} a^{2} + \frac{9}{196} a + \frac{2}{49}$, $\frac{1}{293021143949483108799948688900272522093748923635866698439279852928136} a^{17} - \frac{1046893286173235868792230997223034035084798475717912967960119775}{12209214331228462866664528704178021753906205151494445768303327205339} a^{16} - \frac{541381392583803831454848957739631939318834552145213546028816588757}{97673714649827702933316229633424174031249641211955566146426617642712} a^{15} + \frac{1057278673319594055787790614846853263050376709565189736202068644135}{97673714649827702933316229633424174031249641211955566146426617642712} a^{14} - \frac{2880021246012361032727447418628417298576663652373278770555287534791}{13953387807118243276188032804774882004464234458850795163775231091816} a^{13} - \frac{1177181802048827691431766144203307547301734795036881332595876943439}{13953387807118243276188032804774882004464234458850795163775231091816} a^{12} + \frac{42950570965928296743478923229578202847234131055533726008972451352}{747502918238477318367216043112940107382012560295578312345101665633} a^{11} + \frac{658465941909385780856628655902795830368570713300783092770030684181}{1993341115302606182312576114967840286352033494121542166253604441688} a^{10} + \frac{2791231321237942621456745582868264551693308669282543729643472186709}{6976693903559121638094016402387441002232117229425397581887615545908} a^{9} + \frac{9350749039050752921731870259557034353825616960131527680266885081645}{20930081710677364914282049207162323006696351688276192745662846637724} a^{8} - \frac{249590773716451385722751839969634873924964898134505663644335985585}{996670557651303091156288057483920143176016747060771083126802220844} a^{7} + \frac{1545296024059168371491095637194137679742854010904585121286888423369}{3488346951779560819047008201193720501116058614712698790943807772954} a^{6} + \frac{3002523176683564302850384960994526557966380954344052269730777931353}{6976693903559121638094016402387441002232117229425397581887615545908} a^{5} - \frac{671617141204764482692008081936736493622547396196657294285091155969}{6976693903559121638094016402387441002232117229425397581887615545908} a^{4} - \frac{5862188696231604149802223851211483278087304340822130203704778344171}{24418428662456925733329057408356043507812410302988891536606654410678} a^{3} - \frac{9762295488463224509309800434533443988441588127801461779056289153361}{48836857324913851466658114816712087015624820605977783073213308821356} a^{2} + \frac{4576691203305522128206719575402662991997216899203815965567072284465}{24418428662456925733329057408356043507812410302988891536606654410678} a + \frac{1126536566507562508937779773099242380292497017809908542532945399396}{12209214331228462866664528704178021753906205151494445768303327205339}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 175948770443000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{18}$ (as 18T982):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3201186852864000
The 200 conjugacy class representatives for $A_{18}$ are not computed
Character table for $A_{18}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

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 R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.11.0.1}{11} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.13.0.1}{13} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.8.22.95$x^{8} + 184 x^{4} + 400$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.8.10$x^{6} + 6 x^{5} + 36$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.10.10.7$x^{10} + 10 x^{8} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 12$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.10.9.2$x^{10} + 14$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$