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Properties

Label 18.0.51763447257...4288.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{9}\cdot 19^{6}\cdot 97^{6}$
Root discriminant $57.79$
Ramified primes $2, 3, 19, 97$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![703, 1900, 25906, 71956, 3110, -100260, 19897, 30174, -15438, -686, 7298, -5022, 2135, -714, 216, -78, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 24*x^16 - 78*x^15 + 216*x^14 - 714*x^13 + 2135*x^12 - 5022*x^11 + 7298*x^10 - 686*x^9 - 15438*x^8 + 30174*x^7 + 19897*x^6 - 100260*x^5 + 3110*x^4 + 71956*x^3 + 25906*x^2 + 1900*x + 703)
 
gp: K = bnfinit(x^18 - 6*x^17 + 24*x^16 - 78*x^15 + 216*x^14 - 714*x^13 + 2135*x^12 - 5022*x^11 + 7298*x^10 - 686*x^9 - 15438*x^8 + 30174*x^7 + 19897*x^6 - 100260*x^5 + 3110*x^4 + 71956*x^3 + 25906*x^2 + 1900*x + 703, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 24 x^{16} - 78 x^{15} + 216 x^{14} - 714 x^{13} + 2135 x^{12} - 5022 x^{11} + 7298 x^{10} - 686 x^{9} - 15438 x^{8} + 30174 x^{7} + 19897 x^{6} - 100260 x^{5} + 3110 x^{4} + 71956 x^{3} + 25906 x^{2} + 1900 x + 703 \)

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:  \(-51763447257109020902974751244288=-\,2^{26}\cdot 3^{9}\cdot 19^{6}\cdot 97^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.79$
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:  $2, 3, 19, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{1394186} a^{15} - \frac{5}{1394186} a^{14} + \frac{90849}{697093} a^{13} - \frac{133493}{697093} a^{12} + \frac{47292}{697093} a^{11} - \frac{101134}{697093} a^{10} - \frac{158887}{1394186} a^{9} + \frac{239339}{697093} a^{8} - \frac{25698}{697093} a^{7} + \frac{5529}{1394186} a^{6} + \frac{235683}{697093} a^{5} + \frac{315487}{697093} a^{4} + \frac{325819}{697093} a^{3} + \frac{323855}{1394186} a^{2} + \frac{2815}{697093} a + \frac{128797}{1394186}$, $\frac{1}{9759302} a^{16} - \frac{3}{9759302} a^{15} + \frac{2272967}{9759302} a^{14} + \frac{745298}{4879651} a^{13} + \frac{1651891}{9759302} a^{12} - \frac{703643}{4879651} a^{11} + \frac{66835}{4879651} a^{10} - \frac{1930375}{9759302} a^{9} - \frac{2335392}{4879651} a^{8} + \frac{1694101}{4879651} a^{7} - \frac{1608855}{9759302} a^{6} - \frac{1304426}{4879651} a^{5} + \frac{1216493}{9759302} a^{4} - \frac{2323353}{4879651} a^{3} + \frac{192919}{1394186} a^{2} - \frac{139373}{697093} a - \frac{1962482}{4879651}$, $\frac{1}{62224296667317392782221633362} a^{17} - \frac{1038722506903870908382}{31112148333658696391110816681} a^{16} - \frac{1307186826390701969336}{31112148333658696391110816681} a^{15} - \frac{4103052004441625500946259714}{31112148333658696391110816681} a^{14} - \frac{320103014509088679793811874}{4444592619094099484444402383} a^{13} - \frac{92473624916120079222493761}{1269883605455456995555543538} a^{12} + \frac{249218035709470649402287485}{4444592619094099484444402383} a^{11} - \frac{2425688252811691288638168320}{31112148333658696391110816681} a^{10} - \frac{11260645771388281437097127385}{62224296667317392782221633362} a^{9} - \frac{759839738513099838681838193}{1637481491245194546900569299} a^{8} + \frac{11300082473788638202638268458}{31112148333658696391110816681} a^{7} - \frac{1803655089198057831642400223}{8889185238188198968888804766} a^{6} + \frac{13651570873046504303658737225}{62224296667317392782221633362} a^{5} - \frac{2230634794512788962820769946}{31112148333658696391110816681} a^{4} - \frac{11852252024752839402875865387}{62224296667317392782221633362} a^{3} - \frac{2090362849009697891276647991}{4444592619094099484444402383} a^{2} + \frac{109640209214640600737948940}{1637481491245194546900569299} a + \frac{204445605171525418193809502}{1637481491245194546900569299}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{28079280852}{89014065842599} a^{17} + \frac{175872361889}{89014065842599} a^{16} - \frac{1442240101377}{178028131685198} a^{15} + \frac{4777022621963}{178028131685198} a^{14} - \frac{6731116398720}{89014065842599} a^{13} + \frac{21952196886656}{89014065842599} a^{12} - \frac{132238047001901}{178028131685198} a^{11} + \frac{319183727878739}{178028131685198} a^{10} - \frac{501354972398623}{178028131685198} a^{9} + \frac{9991909927137}{9369901667642} a^{8} + \frac{780433159484867}{178028131685198} a^{7} - \frac{1869158341439505}{178028131685198} a^{6} - \frac{587882334282259}{178028131685198} a^{5} + \frac{5650489807077083}{178028131685198} a^{4} - \frac{768664550714144}{89014065842599} a^{3} - \frac{1647153017172606}{89014065842599} a^{2} - \frac{50871716161445}{9369901667642} a - \frac{99236886313}{9369901667642} \) (order $6$)
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:  \( 511066585.49785817 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.152.1, 6.0.4064688.2, 6.0.623808.1

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

Sibling fields

Degree 12 sibling: data not computed
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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.12.22.27$x^{12} + 2 x^{6} + 4$$6$$2$$22$$C_6\times S_3$$[3]_{3}^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$97$97.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
97.3.2.2$x^{3} + 485$$3$$1$$2$$C_3$$[\ ]_{3}$
97.6.4.3$x^{6} + 873 x^{3} + 235225$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
97.6.0.1$x^{6} - x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$