↑

Properties

Label 18.8.15191184967...9351.3
Degree $18$
Signature $[8, 5]$
Discriminant $-\,7^{12}\cdot 97^{2}\cdot 22679^{3}$
Root discriminant $32.37$
Ramified primes $7, 97, 22679$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T840

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![559, -42537, -39498, 80746, 29206, -62804, -12276, 30868, 4023, -11479, -561, 3076, -18, -642, 65, 82, -16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 16*x^16 + 82*x^15 + 65*x^14 - 642*x^13 - 18*x^12 + 3076*x^11 - 561*x^10 - 11479*x^9 + 4023*x^8 + 30868*x^7 - 12276*x^6 - 62804*x^5 + 29206*x^4 + 80746*x^3 - 39498*x^2 - 42537*x + 559)
 
gp: K = bnfinit(x^18 - 4*x^17 - 16*x^16 + 82*x^15 + 65*x^14 - 642*x^13 - 18*x^12 + 3076*x^11 - 561*x^10 - 11479*x^9 + 4023*x^8 + 30868*x^7 - 12276*x^6 - 62804*x^5 + 29206*x^4 + 80746*x^3 - 39498*x^2 - 42537*x + 559, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 16 x^{16} + 82 x^{15} + 65 x^{14} - 642 x^{13} - 18 x^{12} + 3076 x^{11} - 561 x^{10} - 11479 x^{9} + 4023 x^{8} + 30868 x^{7} - 12276 x^{6} - 62804 x^{5} + 29206 x^{4} + 80746 x^{3} - 39498 x^{2} - 42537 x + 559 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1519118496790264747045229351=-\,7^{12}\cdot 97^{2}\cdot 22679^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.37$
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:  $7, 97, 22679$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{113} a^{16} + \frac{7}{113} a^{15} + \frac{41}{113} a^{14} + \frac{54}{113} a^{13} - \frac{48}{113} a^{12} + \frac{10}{113} a^{11} + \frac{35}{113} a^{10} - \frac{16}{113} a^{9} + \frac{32}{113} a^{8} + \frac{41}{113} a^{7} - \frac{8}{113} a^{6} + \frac{15}{113} a^{5} + \frac{27}{113} a^{4} + \frac{21}{113} a^{3} - \frac{31}{113} a^{2} - \frac{19}{113} a + \frac{11}{113}$, $\frac{1}{4376768144029355111433375631232684881} a^{17} + \frac{14354412929352108446946757513607269}{4376768144029355111433375631232684881} a^{16} - \frac{1171844064423165312957040813598007336}{4376768144029355111433375631232684881} a^{15} - \frac{1171692340987032552467100657397530305}{4376768144029355111433375631232684881} a^{14} - \frac{1117257548421388839243540379359856337}{4376768144029355111433375631232684881} a^{13} - \frac{995170806496427015563061584492494763}{4376768144029355111433375631232684881} a^{12} - \frac{836331973213364863669401005215021416}{4376768144029355111433375631232684881} a^{11} + \frac{1828246011952728949507676268867428850}{4376768144029355111433375631232684881} a^{10} + \frac{629091216471526855780267633362002136}{4376768144029355111433375631232684881} a^{9} + \frac{1932402899891124138539464056624536823}{4376768144029355111433375631232684881} a^{8} + \frac{1782788447935443730428205285233166606}{4376768144029355111433375631232684881} a^{7} - \frac{1526243984285977698212305880097569985}{4376768144029355111433375631232684881} a^{6} - \frac{98752537202497282570845188074184146}{4376768144029355111433375631232684881} a^{5} - \frac{448020452144997997344658484025957817}{4376768144029355111433375631232684881} a^{4} - \frac{1623623007357363051516748024795587241}{4376768144029355111433375631232684881} a^{3} - \frac{1791805056416217361888671672056817240}{4376768144029355111433375631232684881} a^{2} - \frac{1845817798844822194636217994868563079}{4376768144029355111433375631232684881} a + \frac{55522461589160853084566137563737011}{4376768144029355111433375631232684881}$

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:  $12$
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:  \( 3254034.77066 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T840:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 180 conjugacy class representatives for t18n840 are not computed
Character table for t18n840 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.7.2668161671.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$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}$
97Data not computed
22679Data not computed