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Properties

Label 18.10.1353065020...5424.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}$
Root discriminant $53.64$
Ramified primes $2, 3, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18509, 51566, -2336, -119096, 123525, -1652, -74449, 71380, -48195, 15362, 14670, -19220, 7776, -102, -925, 268, -7, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 7*x^16 + 268*x^15 - 925*x^14 - 102*x^13 + 7776*x^12 - 19220*x^11 + 14670*x^10 + 15362*x^9 - 48195*x^8 + 71380*x^7 - 74449*x^6 - 1652*x^5 + 123525*x^4 - 119096*x^3 - 2336*x^2 + 51566*x - 18509)
 
gp: K = bnfinit(x^18 - 8*x^17 - 7*x^16 + 268*x^15 - 925*x^14 - 102*x^13 + 7776*x^12 - 19220*x^11 + 14670*x^10 + 15362*x^9 - 48195*x^8 + 71380*x^7 - 74449*x^6 - 1652*x^5 + 123525*x^4 - 119096*x^3 - 2336*x^2 + 51566*x - 18509, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 7 x^{16} + 268 x^{15} - 925 x^{14} - 102 x^{13} + 7776 x^{12} - 19220 x^{11} + 14670 x^{10} + 15362 x^{9} - 48195 x^{8} + 71380 x^{7} - 74449 x^{6} - 1652 x^{5} + 123525 x^{4} - 119096 x^{3} - 2336 x^{2} + 51566 x - 18509 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13530650209602087833780282855424=2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.64$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{9} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{5}{12}$, $\frac{1}{12} a^{13} - \frac{1}{6} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{5}{12} a$, $\frac{1}{12} a^{14} - \frac{1}{6} a^{11} + \frac{1}{12} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{5}{12} a^{2} - \frac{1}{2}$, $\frac{1}{168} a^{15} - \frac{5}{168} a^{14} - \frac{1}{42} a^{13} - \frac{1}{168} a^{12} - \frac{25}{168} a^{11} + \frac{25}{168} a^{10} + \frac{3}{14} a^{9} + \frac{17}{168} a^{8} - \frac{17}{84} a^{7} + \frac{17}{84} a^{6} + \frac{5}{14} a^{5} - \frac{23}{84} a^{4} - \frac{1}{168} a^{3} + \frac{41}{168} a^{2} - \frac{8}{21} a + \frac{1}{168}$, $\frac{1}{168} a^{16} - \frac{1}{168} a^{14} - \frac{1}{24} a^{13} - \frac{1}{84} a^{12} + \frac{1}{14} a^{11} - \frac{1}{24} a^{10} + \frac{5}{56} a^{9} + \frac{23}{168} a^{8} + \frac{1}{42} a^{7} + \frac{17}{84} a^{6} - \frac{13}{84} a^{5} + \frac{11}{24} a^{4} - \frac{5}{42} a^{3} + \frac{29}{168} a^{2} + \frac{31}{168} a + \frac{11}{56}$, $\frac{1}{31583504191114777111479528} a^{17} - \frac{12992023202342193670877}{7895876047778694277869882} a^{16} - \frac{22713063588728495762875}{10527834730371592370493176} a^{15} + \frac{443776065571662532043785}{31583504191114777111479528} a^{14} - \frac{32296961835992824477801}{1127982292539813468267126} a^{13} + \frac{425467500034296438999347}{15791752095557388555739764} a^{12} + \frac{2306281491587299057365287}{10527834730371592370493176} a^{11} - \frac{5429398560664525223072605}{31583504191114777111479528} a^{10} - \frac{7029755538600323196620503}{31583504191114777111479528} a^{9} + \frac{419832370897640568131675}{2255964585079626936534252} a^{8} + \frac{2352209847785783046598231}{15791752095557388555739764} a^{7} + \frac{2387292133661827736754229}{15791752095557388555739764} a^{6} - \frac{10301777782327851275278523}{31583504191114777111479528} a^{5} - \frac{686905912945321373454641}{7895876047778694277869882} a^{4} + \frac{12933562334013900909175393}{31583504191114777111479528} a^{3} - \frac{1100897504109979841492713}{31583504191114777111479528} a^{2} + \frac{553799546009306182691367}{1503976390053084624356168} a - \frac{5152231889761728889828073}{15791752095557388555739764}$

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:  $13$
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:  \( 1715110594.83 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$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}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.6.5.2$x^{6} + 246$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$