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Properties

Label 18.0.70530723208...7031.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{12}\cdot 71^{5}$
Root discriminant $51.74$
Ramified primes $3, 7, 71$
Class number $1360$ (GRH)
Class group $[2, 2, 2, 170]$ (GRH)
Galois group $C_2\times A_4^2$ (as 18T109)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![112589, -168924, 931392, 1332753, 1683297, 1108374, 847530, 417480, 231111, 82138, 44889, 6999, 7003, -195, 768, -70, 45, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 45*x^16 - 70*x^15 + 768*x^14 - 195*x^13 + 7003*x^12 + 6999*x^11 + 44889*x^10 + 82138*x^9 + 231111*x^8 + 417480*x^7 + 847530*x^6 + 1108374*x^5 + 1683297*x^4 + 1332753*x^3 + 931392*x^2 - 168924*x + 112589)
 
gp: K = bnfinit(x^18 - 3*x^17 + 45*x^16 - 70*x^15 + 768*x^14 - 195*x^13 + 7003*x^12 + 6999*x^11 + 44889*x^10 + 82138*x^9 + 231111*x^8 + 417480*x^7 + 847530*x^6 + 1108374*x^5 + 1683297*x^4 + 1332753*x^3 + 931392*x^2 - 168924*x + 112589, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 45 x^{16} - 70 x^{15} + 768 x^{14} - 195 x^{13} + 7003 x^{12} + 6999 x^{11} + 44889 x^{10} + 82138 x^{9} + 231111 x^{8} + 417480 x^{7} + 847530 x^{6} + 1108374 x^{5} + 1683297 x^{4} + 1332753 x^{3} + 931392 x^{2} - 168924 x + 112589 \)

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:  \(-7053072320828130442597556717031=-\,3^{24}\cdot 7^{12}\cdot 71^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.74$
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:  $3, 7, 71$
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}{390093678645019159048399035309778602971092439474809} a^{17} + \frac{165094809400759974685110439511391240345597836837561}{390093678645019159048399035309778602971092439474809} a^{16} + \frac{69489281613448268037415265791131136366585680859953}{390093678645019159048399035309778602971092439474809} a^{15} - \frac{161073942860686408410104721535649812937992998230227}{390093678645019159048399035309778602971092439474809} a^{14} - \frac{175435984300104882267833977252722173264188714796340}{390093678645019159048399035309778602971092439474809} a^{13} - \frac{97124335933749902456832743465163389395704818313996}{390093678645019159048399035309778602971092439474809} a^{12} + \frac{114886620766730882828423128710619938974398794266800}{390093678645019159048399035309778602971092439474809} a^{11} + \frac{121574453173592987880631224697197967548892484751139}{390093678645019159048399035309778602971092439474809} a^{10} + \frac{171285719690957602073172321796840437858950264624373}{390093678645019159048399035309778602971092439474809} a^{9} - \frac{2671056587832049584015891611960282752733800683217}{390093678645019159048399035309778602971092439474809} a^{8} + \frac{26907513423359663115846552928898728514077861460161}{390093678645019159048399035309778602971092439474809} a^{7} + \frac{108651037073796508140024550619910022787014326088847}{390093678645019159048399035309778602971092439474809} a^{6} - \frac{142720114408560143274035645932565869535213438358303}{390093678645019159048399035309778602971092439474809} a^{5} + \frac{27221271043321912411669508716717552844428533672791}{390093678645019159048399035309778602971092439474809} a^{4} - \frac{109342023339605431344090076985466580782657318380049}{390093678645019159048399035309778602971092439474809} a^{3} + \frac{29103213523277467049568410086174932124507710535347}{390093678645019159048399035309778602971092439474809} a^{2} - \frac{165513758322712583648966591971058862518721952587566}{390093678645019159048399035309778602971092439474809} a - \frac{101649908437883463671610329250028578614100335713415}{390093678645019159048399035309778602971092439474809}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{170}$, which has order $1360$ (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:  \( 54408.4888887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_4^2$ (as 18T109):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 32 conjugacy class representatives for $C_2\times A_4^2$
Character table for $C_2\times A_4^2$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{9})^+\), 9.9.62523502209.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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}$
7Data not computed
71Data not computed