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Properties

Label 18.6.12178167588...9257.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 73^{3}\cdot 577^{4}$
Root discriminant $36.33$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1917, -5670, -3213, 7515, 15120, 15840, 12504, 7110, 1770, -115, 729, 207, -334, -42, -36, -31, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 31*x^15 - 36*x^14 - 42*x^13 - 334*x^12 + 207*x^11 + 729*x^10 - 115*x^9 + 1770*x^8 + 7110*x^7 + 12504*x^6 + 15840*x^5 + 15120*x^4 + 7515*x^3 - 3213*x^2 - 5670*x - 1917)
 
gp: K = bnfinit(x^18 + 9*x^16 - 31*x^15 - 36*x^14 - 42*x^13 - 334*x^12 + 207*x^11 + 729*x^10 - 115*x^9 + 1770*x^8 + 7110*x^7 + 12504*x^6 + 15840*x^5 + 15120*x^4 + 7515*x^3 - 3213*x^2 - 5670*x - 1917, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 31 x^{15} - 36 x^{14} - 42 x^{13} - 334 x^{12} + 207 x^{11} + 729 x^{10} - 115 x^{9} + 1770 x^{8} + 7110 x^{7} + 12504 x^{6} + 15840 x^{5} + 15120 x^{4} + 7515 x^{3} - 3213 x^{2} - 5670 x - 1917 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12178167588536804870357659257=3^{24}\cdot 73^{3}\cdot 577^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.33$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} + \frac{1}{3} a^{10} - \frac{4}{9} a^{9} - \frac{1}{9} a^{6} - \frac{1}{3} a^{5}$, $\frac{1}{171} a^{16} + \frac{1}{57} a^{15} - \frac{8}{57} a^{14} - \frac{1}{171} a^{13} - \frac{49}{171} a^{10} + \frac{1}{57} a^{9} + \frac{20}{57} a^{8} + \frac{53}{171} a^{7} + \frac{4}{19} a^{6} + \frac{11}{57} a^{5} - \frac{9}{19} a^{4} - \frac{13}{57} a^{3} - \frac{7}{19} a^{2} + \frac{8}{19} a - \frac{1}{19}$, $\frac{1}{652503400061917536792348826707} a^{17} - \frac{920805418773135883270326536}{652503400061917536792348826707} a^{16} - \frac{999579662374350285313943934}{72500377784657504088038758523} a^{15} - \frac{26360846270709806803208515831}{652503400061917536792348826707} a^{14} + \frac{57990451141594138545504003893}{652503400061917536792348826707} a^{13} + \frac{1268603697181409079046226614}{11447428071261711171795593451} a^{12} - \frac{5151144828622050820422794512}{652503400061917536792348826707} a^{11} + \frac{220190640674691164357903483393}{652503400061917536792348826707} a^{10} - \frac{8007519593926768471968936263}{72500377784657504088038758523} a^{9} + \frac{28410708193626957713581332680}{652503400061917536792348826707} a^{8} + \frac{257630929185149042463648013598}{652503400061917536792348826707} a^{7} - \frac{68743936604519950020548462965}{217501133353972512264116275569} a^{6} - \frac{11120176297584031539588167202}{72500377784657504088038758523} a^{5} + \frac{56530474343770452528238311034}{217501133353972512264116275569} a^{4} - \frac{20536403192518462473715950757}{72500377784657504088038758523} a^{3} - \frac{1699942264361978361325363203}{3815809357087237057265197817} a^{2} - \frac{966906160455855827319146649}{3815809357087237057265197817} a - \frac{17548999912346302766144739847}{72500377784657504088038758523}$

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:  $11$
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:  \( 9911769.96066 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
73Data not computed
577Data not computed