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Properties

Label 18.0.11759452462...4032.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{18}\cdot 7^{15}\cdot 29^{3}$
Root discriminant $53.23$
Ramified primes $2, 3, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T857

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1560896, -5005632, 8598384, -9426624, 8297016, -5763924, 3719465, -1863372, 964062, -348920, 165189, -36612, 19430, -1944, 1458, -40, 60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 60*x^16 - 40*x^15 + 1458*x^14 - 1944*x^13 + 19430*x^12 - 36612*x^11 + 165189*x^10 - 348920*x^9 + 964062*x^8 - 1863372*x^7 + 3719465*x^6 - 5763924*x^5 + 8297016*x^4 - 9426624*x^3 + 8598384*x^2 - 5005632*x + 1560896)
 
gp: K = bnfinit(x^18 + 60*x^16 - 40*x^15 + 1458*x^14 - 1944*x^13 + 19430*x^12 - 36612*x^11 + 165189*x^10 - 348920*x^9 + 964062*x^8 - 1863372*x^7 + 3719465*x^6 - 5763924*x^5 + 8297016*x^4 - 9426624*x^3 + 8598384*x^2 - 5005632*x + 1560896, 1)
 

Normalized defining polynomial

\( x^{18} + 60 x^{16} - 40 x^{15} + 1458 x^{14} - 1944 x^{13} + 19430 x^{12} - 36612 x^{11} + 165189 x^{10} - 348920 x^{9} + 964062 x^{8} - 1863372 x^{7} + 3719465 x^{6} - 5763924 x^{5} + 8297016 x^{4} - 9426624 x^{3} + 8598384 x^{2} - 5005632 x + 1560896 \)

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:  \(-11759452462469462530524552364032=-\,2^{18}\cdot 3^{18}\cdot 7^{15}\cdot 29^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.23$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{464} a^{15} - \frac{7}{58} a^{13} - \frac{5}{58} a^{12} - \frac{25}{232} a^{11} + \frac{7}{116} a^{10} - \frac{1}{8} a^{9} + \frac{11}{116} a^{8} - \frac{111}{464} a^{7} - \frac{27}{116} a^{6} + \frac{51}{232} a^{5} - \frac{4}{29} a^{4} + \frac{41}{464} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{928} a^{16} - \frac{7}{116} a^{14} - \frac{5}{116} a^{13} + \frac{33}{464} a^{12} - \frac{11}{116} a^{11} - \frac{1}{16} a^{10} + \frac{11}{232} a^{9} + \frac{5}{928} a^{8} - \frac{7}{29} a^{7} - \frac{7}{464} a^{6} + \frac{13}{232} a^{5} + \frac{273}{928} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{64805016604476289296835661837261711936} a^{17} - \frac{11165466613021575531421870095569}{69832992030685656569866014910842362} a^{16} + \frac{2841118034334960230315724577314977}{8100627075559536162104457729657713992} a^{15} - \frac{37853262204569053570763195218238460}{1012578384444942020263057216207214249} a^{14} + \frac{1761959713623030675561752782088583233}{32402508302238144648417830918630855968} a^{13} - \frac{354211382708949062533349343912207439}{8100627075559536162104457729657713992} a^{12} - \frac{27202460399321838180119814526072045}{1117327872490970505117856238573477792} a^{11} - \frac{108715267061640795460397406516740973}{16201254151119072324208915459315427984} a^{10} + \frac{7548879153607341145892900462289465725}{64805016604476289296835661837261711936} a^{9} - \frac{558844815157966525619859964811806695}{8100627075559536162104457729657713992} a^{8} + \frac{1909551474984851240418374989198751805}{32402508302238144648417830918630855968} a^{7} - \frac{3650867208599216970595384531299486409}{16201254151119072324208915459315427984} a^{6} + \frac{28602188859365228738023375380493867345}{64805016604476289296835661837261711936} a^{5} - \frac{140437518567800073491798707989261977}{558663936245485252558928119286738896} a^{4} + \frac{245956631309296712838161680150040741}{558663936245485252558928119286738896} a^{3} + \frac{92738894379130717722196122476997083}{279331968122742626279464059643369448} a^{2} + \frac{542084510999565122100465264232388}{1204017103977338906377000257083489} a - \frac{582065907916101349535111500004708}{1204017103977338906377000257083489}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{42639449198373999986293971855}{69832992030685656569866014910842362} a^{17} + \frac{816316713092285134115210569899}{1117327872490970505117856238573477792} a^{16} - \frac{15788985103421635249055425945037}{558663936245485252558928119286738896} a^{15} + \frac{8124745645093236156639540941823}{139665984061371313139732029821684724} a^{14} - \frac{33371932673078534472138570009033}{69832992030685656569866014910842362} a^{13} + \frac{774869289431173966480982639357131}{558663936245485252558928119286738896} a^{12} - \frac{1139675421737719523983947677347851}{279331968122742626279464059643369448} a^{11} + \frac{6566350349544145027543995492208317}{558663936245485252558928119286738896} a^{10} - \frac{1602850042071840637674577710856041}{69832992030685656569866014910842362} a^{9} + \frac{10939641303574977023386536858744615}{1117327872490970505117856238573477792} a^{8} - \frac{38854047747877583534215903434052629}{558663936245485252558928119286738896} a^{7} - \frac{167294381061989399636926188846084169}{558663936245485252558928119286738896} a^{6} + \frac{902680106682115419056872729852365}{4816068415909355625508001028333956} a^{5} - \frac{50210399499254927991628965383692161}{38528547327274845004064008226671648} a^{4} + \frac{709301424773104268338358140660044537}{558663936245485252558928119286738896} a^{3} - \frac{20837340264156651350551861510582503}{9632136831818711251016002056667912} a^{2} + \frac{352484035633233516218609317724685}{2408034207954677812754000514166978} a + \frac{840476940957241698571738350819013}{2408034207954677812754000514166978} \) (order $14$)
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:  \( 1037490994.38 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T857:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 159 conjugacy class representatives for t18n857 are not computed
Character table for t18n857 is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\)

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.6$x^{6} + 4 x^{2} + 8$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
7Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$