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Properties

Label 18.0.45198561639...3584.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{6}\cdot 3^{24}\cdot 13^{12}\cdot 181^{4}$
Root discriminant $95.68$
Ramified primes $2, 3, 13, 181$
Class number $56208$ (GRH)
Class group $[2, 2, 2, 7026]$ (GRH)
Galois group 18T459

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45232552, -147873180, 233241462, -232245897, 164458332, -88940514, 39375297, -15415413, 5830770, -2094040, 718602, -210678, 57423, -10842, 2784, -221, 81, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 81*x^16 - 221*x^15 + 2784*x^14 - 10842*x^13 + 57423*x^12 - 210678*x^11 + 718602*x^10 - 2094040*x^9 + 5830770*x^8 - 15415413*x^7 + 39375297*x^6 - 88940514*x^5 + 164458332*x^4 - 232245897*x^3 + 233241462*x^2 - 147873180*x + 45232552)
 
gp: K = bnfinit(x^18 + 81*x^16 - 221*x^15 + 2784*x^14 - 10842*x^13 + 57423*x^12 - 210678*x^11 + 718602*x^10 - 2094040*x^9 + 5830770*x^8 - 15415413*x^7 + 39375297*x^6 - 88940514*x^5 + 164458332*x^4 - 232245897*x^3 + 233241462*x^2 - 147873180*x + 45232552, 1)
 

Normalized defining polynomial

\( x^{18} + 81 x^{16} - 221 x^{15} + 2784 x^{14} - 10842 x^{13} + 57423 x^{12} - 210678 x^{11} + 718602 x^{10} - 2094040 x^{9} + 5830770 x^{8} - 15415413 x^{7} + 39375297 x^{6} - 88940514 x^{5} + 164458332 x^{4} - 232245897 x^{3} + 233241462 x^{2} - 147873180 x + 45232552 \)

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:  \(-451985616395723240517472053274603584=-\,2^{6}\cdot 3^{24}\cdot 13^{12}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.68$
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:  $2, 3, 13, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} + \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{7132610273633665135771863981818595582731243211600709295653423446128} a^{17} - \frac{2013110748278549190331492114934063253654053584434117050365907393}{48853495024888117368300438231634216320077008298634995175708379768} a^{16} + \frac{2267506357994971545257784796021619255185707905280609571664412207253}{7132610273633665135771863981818595582731243211600709295653423446128} a^{15} - \frac{736223267159710285609155835430105398453652673064022156964458967447}{7132610273633665135771863981818595582731243211600709295653423446128} a^{14} - \frac{1302307813507602277337422545388909483988493782678382222592299523817}{3566305136816832567885931990909297791365621605800354647826711723064} a^{13} + \frac{988289242023435913592147241296704288126631303436218316149830805037}{3566305136816832567885931990909297791365621605800354647826711723064} a^{12} - \frac{163238131433183439566793831578458046755746430252899183034307816821}{7132610273633665135771863981818595582731243211600709295653423446128} a^{11} - \frac{11392074104268542934477539485554731609155530705044077668545849615}{24426747512444058684150219115817108160038504149317497587854189884} a^{10} + \frac{624664268302491718184194009839252198392179031128858218167966864937}{3566305136816832567885931990909297791365621605800354647826711723064} a^{9} - \frac{172548957532422022365587984782055651357243926859037266698081904187}{1783152568408416283942965995454648895682810802900177323913355861532} a^{8} - \frac{794724174442899263583005632693928139277353714092884873743778627515}{3566305136816832567885931990909297791365621605800354647826711723064} a^{7} + \frac{1083455840038401439433628875509086454742105333927490122158950845175}{7132610273633665135771863981818595582731243211600709295653423446128} a^{6} - \frac{496823109681959261785353911915787018941572374806473336596887429949}{7132610273633665135771863981818595582731243211600709295653423446128} a^{5} - \frac{359664044360627579835018259860087858867614682641158169052341475595}{891576284204208141971482997727324447841405401450088661956677930766} a^{4} + \frac{364972811924660636344438142831309892195300683858819575602504555027}{1783152568408416283942965995454648895682810802900177323913355861532} a^{3} + \frac{1076517765551327386868703888563174860915284944380068558789477978111}{7132610273633665135771863981818595582731243211600709295653423446128} a^{2} + \frac{311534123778575339195651609533500625812902177511980423954422801487}{891576284204208141971482997727324447841405401450088661956677930766} a - \frac{7673463423532430610989477686477764696076653742415988607685395963}{24426747512444058684150219115817108160038504149317497587854189884}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{7026}$, which has order $56208$ (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:  \( -1 \) (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:  \( 400417.136445 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T459:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 96 conjugacy class representatives for t18n459 are not computed
Character table for t18n459 is not computed

Intermediate fields

3.3.13689.2, 3.3.13689.1, \(\Q(\zeta_{9})^+\), 3.3.169.1, 9.9.2565164201769.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
181Data not computed