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Properties

Label 18.0.32836175077...0983.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 83^{4}\cdot 167^{3}\cdot 181^{4}$
Root discriminant $72.78$
Ramified primes $7, 83, 167, 181$
Class number $2298$ (GRH)
Class group $[2298]$ (GRH)
Galois group 18T400

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![651203, 752759, 1004773, 529829, 425870, 72083, -4129, -9069, -4524, 3048, 8545, 797, -1897, -308, 320, 33, -25, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 25*x^16 + 33*x^15 + 320*x^14 - 308*x^13 - 1897*x^12 + 797*x^11 + 8545*x^10 + 3048*x^9 - 4524*x^8 - 9069*x^7 - 4129*x^6 + 72083*x^5 + 425870*x^4 + 529829*x^3 + 1004773*x^2 + 752759*x + 651203)
 
gp: K = bnfinit(x^18 - 2*x^17 - 25*x^16 + 33*x^15 + 320*x^14 - 308*x^13 - 1897*x^12 + 797*x^11 + 8545*x^10 + 3048*x^9 - 4524*x^8 - 9069*x^7 - 4129*x^6 + 72083*x^5 + 425870*x^4 + 529829*x^3 + 1004773*x^2 + 752759*x + 651203, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 25 x^{16} + 33 x^{15} + 320 x^{14} - 308 x^{13} - 1897 x^{12} + 797 x^{11} + 8545 x^{10} + 3048 x^{9} - 4524 x^{8} - 9069 x^{7} - 4129 x^{6} + 72083 x^{5} + 425870 x^{4} + 529829 x^{3} + 1004773 x^{2} + 752759 x + 651203 \)

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:  \(-3283617507769220970246128004960983=-\,7^{12}\cdot 83^{4}\cdot 167^{3}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.78$
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:  $7, 83, 167, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{13} + \frac{2}{7} a^{12} - \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} - \frac{1}{7} a^{8} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{3}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{13} + \frac{3}{7} a^{10} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{13} - \frac{2}{7} a^{12} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{20784077648123270562629282747707941212165825106089} a^{17} - \frac{1224538832585806028636093289323773589590712917076}{20784077648123270562629282747707941212165825106089} a^{16} + \frac{94892053601050440821761728816900757371718189090}{2969153949731895794661326106815420173166546443727} a^{15} - \frac{1171482451862227726592121345463329159371230023575}{20784077648123270562629282747707941212165825106089} a^{14} + \frac{6199262901589551121037105517445225115307847845886}{20784077648123270562629282747707941212165825106089} a^{13} + \frac{1195308112467389784866138638472278632007455236117}{2969153949731895794661326106815420173166546443727} a^{12} + \frac{3523413184671224455065423154128329312840570571544}{20784077648123270562629282747707941212165825106089} a^{11} + \frac{136869341469928296205277522485041595323334879262}{2969153949731895794661326106815420173166546443727} a^{10} - \frac{2895711596648663183629227186113470064271063330680}{20784077648123270562629282747707941212165825106089} a^{9} - \frac{8975160217456438049502007711656594449088973505377}{20784077648123270562629282747707941212165825106089} a^{8} + \frac{8974649077306783293503862975264046398936594542445}{20784077648123270562629282747707941212165825106089} a^{7} + \frac{2557558593178221577430749159435391001029339647684}{20784077648123270562629282747707941212165825106089} a^{6} - \frac{8206156477555432377007543304006733636517299394494}{20784077648123270562629282747707941212165825106089} a^{5} - \frac{1241220161977077878814690233879774724700186045555}{2969153949731895794661326106815420173166546443727} a^{4} + \frac{2364687277779978206516622988005364180082816897044}{20784077648123270562629282747707941212165825106089} a^{3} + \frac{817071945298070243231121087548091920271422599762}{2969153949731895794661326106815420173166546443727} a^{2} - \frac{97073401525952538593251327436800788838386735744}{2969153949731895794661326106815420173166546443727} a - \frac{11234337857555958224799853834285153487718252157}{72418389017851116942959173336961467638208449847}$

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

Class group and class number

$C_{2298}$, which has order $2298$ (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:  \( 1581190.17434 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T400:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2592
The 56 conjugacy class representatives for t18n400 are not computed
Character table for t18n400 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.400967.1, 9.9.26552265046321.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ $18$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.6.4.1$x^{6} + 415 x^{3} + 55112$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
167Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$