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Properties

Label 18.6.74744567481...0064.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 7^{12}\cdot 29^{6}\cdot 53^{6}$
Root discriminant $67.04$
Ramified primes $2, 7, 29, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T268

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14597064, 57979404, 27433944, 19120631, 8506562, 4178615, -721353, -274772, -269658, -252408, -16850, -28568, -6520, -441, -684, 30, -13, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 13*x^16 + 30*x^15 - 684*x^14 - 441*x^13 - 6520*x^12 - 28568*x^11 - 16850*x^10 - 252408*x^9 - 269658*x^8 - 274772*x^7 - 721353*x^6 + 4178615*x^5 + 8506562*x^4 + 19120631*x^3 + 27433944*x^2 + 57979404*x - 14597064)
 
gp: K = bnfinit(x^18 - 3*x^17 - 13*x^16 + 30*x^15 - 684*x^14 - 441*x^13 - 6520*x^12 - 28568*x^11 - 16850*x^10 - 252408*x^9 - 269658*x^8 - 274772*x^7 - 721353*x^6 + 4178615*x^5 + 8506562*x^4 + 19120631*x^3 + 27433944*x^2 + 57979404*x - 14597064, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 13 x^{16} + 30 x^{15} - 684 x^{14} - 441 x^{13} - 6520 x^{12} - 28568 x^{11} - 16850 x^{10} - 252408 x^{9} - 269658 x^{8} - 274772 x^{7} - 721353 x^{6} + 4178615 x^{5} + 8506562 x^{4} + 19120631 x^{3} + 27433944 x^{2} + 57979404 x - 14597064 \)

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:  \(747445674816688288211936682840064=2^{12}\cdot 7^{12}\cdot 29^{6}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.04$
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:  $2, 7, 29, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{582} a^{16} + \frac{93}{194} a^{15} + \frac{161}{582} a^{14} - \frac{29}{97} a^{13} - \frac{22}{97} a^{12} - \frac{87}{194} a^{11} + \frac{106}{291} a^{10} + \frac{11}{291} a^{9} - \frac{73}{291} a^{8} + \frac{6}{97} a^{7} - \frac{43}{97} a^{6} - \frac{13}{291} a^{5} - \frac{57}{194} a^{4} - \frac{31}{582} a^{3} - \frac{95}{291} a^{2} - \frac{151}{582} a - \frac{34}{97}$, $\frac{1}{36085018393603502305925971325784415984471011783830914730565458121372} a^{17} + \frac{592074189748296992248815511014516055895784778218817439332020263}{4009446488178166922880663480642712887163445753758990525618384235708} a^{16} - \frac{10215504385201573206208211889394491743574663656248362262362064737811}{36085018393603502305925971325784415984471011783830914730565458121372} a^{15} + \frac{93087024116830346054420365103146334403455006499435259352937671684}{1002361622044541730720165870160678221790861438439747631404596058927} a^{14} - \frac{34083622581054858341731244024569685593269641853048947065446276021}{1002361622044541730720165870160678221790861438439747631404596058927} a^{13} - \frac{841046970808609645395808701618740803694943417855519362478306916241}{4009446488178166922880663480642712887163445753758990525618384235708} a^{12} + \frac{237895218228990810160250311997229183694976202203056858723384081519}{18042509196801751152962985662892207992235505891915457365282729060686} a^{11} + \frac{453962477905089626643313289879248737612111460394965544582151963874}{9021254598400875576481492831446103996117752945957728682641364530343} a^{10} - \frac{1716776327792337429355223946757545245720080880114910686988958103743}{18042509196801751152962985662892207992235505891915457365282729060686} a^{9} - \frac{371246743157360003480857590021219861709721690918149494520235793319}{1002361622044541730720165870160678221790861438439747631404596058927} a^{8} - \frac{254542233676070471291848727649651256116423747012220090128844434555}{2004723244089083461440331740321356443581722876879495262809192117854} a^{7} - \frac{549571431653655652478986428214396306602686645514939080161820682122}{9021254598400875576481492831446103996117752945957728682641364530343} a^{6} + \frac{1124896460065245881585416372275665332249057878110606935637798786447}{4009446488178166922880663480642712887163445753758990525618384235708} a^{5} + \frac{14984835330264298114002503662589493483147530770052425352066638149305}{36085018393603502305925971325784415984471011783830914730565458121372} a^{4} + \frac{1911655675874322557765779720958695330546883796878400933364397126390}{9021254598400875576481492831446103996117752945957728682641364530343} a^{3} + \frac{4352757178031756755601581740606704128142833784875399685289549555695}{36085018393603502305925971325784415984471011783830914730565458121372} a^{2} - \frac{1640331085902856928709950536447075727591310624710732509845824140709}{6014169732267250384320995220964069330745168630638485788427576353562} a + \frac{78959312977692832228626239688860951602094034506482706560207976034}{1002361622044541730720165870160678221790861438439747631404596058927}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1629568268.27 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T268:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 24 conjugacy class representatives for t18n268
Character table for t18n268 is not computed

Intermediate fields

3.3.2597.1, \(\Q(\zeta_{7})^+\), 9.9.17515230173.1

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.10$x^{12} - 6 x^{10} + 23 x^{8} - 28 x^{6} - 9 x^{4} - 30 x^{2} - 15$$2$$6$$12$12T58$[2, 2, 2, 2]^{6}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$29$29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$