↑

Properties

Label 18.0.47462534282...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{9}\cdot 7^{15}\cdot 13^{15}$
Root discriminant $95.94$
Ramified primes $5, 7, 13$
Class number $153920$ (GRH)
Class group $[2, 2, 2, 2, 2, 4810]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![190444864, 69234848, 104448324, 36129351, 41497328, 3825443, 11950469, -2195492, 2260520, -608429, 359244, -69879, 46893, -3989, 3454, -108, 103, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 103*x^16 - 108*x^15 + 3454*x^14 - 3989*x^13 + 46893*x^12 - 69879*x^11 + 359244*x^10 - 608429*x^9 + 2260520*x^8 - 2195492*x^7 + 11950469*x^6 + 3825443*x^5 + 41497328*x^4 + 36129351*x^3 + 104448324*x^2 + 69234848*x + 190444864)
 
gp: K = bnfinit(x^18 - x^17 + 103*x^16 - 108*x^15 + 3454*x^14 - 3989*x^13 + 46893*x^12 - 69879*x^11 + 359244*x^10 - 608429*x^9 + 2260520*x^8 - 2195492*x^7 + 11950469*x^6 + 3825443*x^5 + 41497328*x^4 + 36129351*x^3 + 104448324*x^2 + 69234848*x + 190444864, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 103 x^{16} - 108 x^{15} + 3454 x^{14} - 3989 x^{13} + 46893 x^{12} - 69879 x^{11} + 359244 x^{10} - 608429 x^{9} + 2260520 x^{8} - 2195492 x^{7} + 11950469 x^{6} + 3825443 x^{5} + 41497328 x^{4} + 36129351 x^{3} + 104448324 x^{2} + 69234848 x + 190444864 \)

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:  \(-474625342823745253277656048537109375=-\,5^{9}\cdot 7^{15}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.94$
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:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(386,·)$, $\chi_{455}(69,·)$, $\chi_{455}(326,·)$, $\chi_{455}(129,·)$, $\chi_{455}(264,·)$, $\chi_{455}(194,·)$, $\chi_{455}(16,·)$, $\chi_{455}(81,·)$, $\chi_{455}(211,·)$, $\chi_{455}(261,·)$, $\chi_{455}(454,·)$, $\chi_{455}(199,·)$, $\chi_{455}(244,·)$, $\chi_{455}(374,·)$, $\chi_{455}(439,·)$, $\chi_{455}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2}$, $\frac{1}{48} a^{15} + \frac{1}{48} a^{14} - \frac{1}{12} a^{13} - \frac{5}{48} a^{12} - \frac{1}{12} a^{11} - \frac{1}{4} a^{10} + \frac{5}{48} a^{9} - \frac{3}{16} a^{8} - \frac{23}{48} a^{7} + \frac{7}{24} a^{6} - \frac{13}{48} a^{5} + \frac{1}{3} a^{4} + \frac{11}{24} a^{3} + \frac{11}{48} a^{2} - \frac{1}{3}$, $\frac{1}{110458032} a^{16} - \frac{259261}{55229016} a^{15} + \frac{741431}{110458032} a^{14} - \frac{2403113}{110458032} a^{13} - \frac{5634841}{110458032} a^{12} - \frac{2873307}{18409672} a^{11} - \frac{17917333}{110458032} a^{10} - \frac{3898545}{18409672} a^{9} - \frac{4995713}{27614508} a^{8} - \frac{2611279}{110458032} a^{7} + \frac{11819153}{110458032} a^{6} + \frac{21074071}{110458032} a^{5} + \frac{874330}{6903627} a^{4} - \frac{29441233}{110458032} a^{3} - \frac{106293}{36819344} a^{2} + \frac{8669627}{27614508} a + \frac{634969}{2301209}$, $\frac{1}{1702981492951204423222354742086763739484141893058649985706272} a^{17} - \frac{338900390650415920431240036976738155238921650297403}{283830248825200737203725790347793956580690315509774997617712} a^{16} - \frac{8920933688953468729062401775996305384619936305236019159323}{1702981492951204423222354742086763739484141893058649985706272} a^{15} - \frac{19196993903553905131633004342141715854290470084394895390823}{567660497650401474407451580695587913161380631019549995235424} a^{14} + \frac{21477601374893148417229026787977271625146428205492899997193}{567660497650401474407451580695587913161380631019549995235424} a^{13} - \frac{17101202248566168588251919212877353216040780502005152235973}{425745373237801105805588685521690934871035473264662496426568} a^{12} - \frac{19700067615589985164201191578004221385998035593217703017871}{1702981492951204423222354742086763739484141893058649985706272} a^{11} - \frac{1295801071419533791813391421111860710745069690654565535667}{106436343309450276451397171380422733717758868316165624106642} a^{10} + \frac{10817243735077905342587319658588918592911612387925934276961}{70957562206300184300931447586948489145172578877443749404428} a^{9} + \frac{200712439790508818369402554395364858090761854827789210437055}{1702981492951204423222354742086763739484141893058649985706272} a^{8} + \frac{649255246694641397048837961512878467347274918670332932763421}{1702981492951204423222354742086763739484141893058649985706272} a^{7} + \frac{208909686532326318030576166044384247715357745184380836756143}{1702981492951204423222354742086763739484141893058649985706272} a^{6} + \frac{107330488985101637116405414486484113546072442869032887134585}{851490746475602211611177371043381869742070946529324992853136} a^{5} + \frac{394166069784980651954353346305911929585216664572440695407681}{1702981492951204423222354742086763739484141893058649985706272} a^{4} + \frac{766549026272553252642102199566174326406599947853013435449567}{1702981492951204423222354742086763739484141893058649985706272} a^{3} + \frac{130473511139037816303907152615748101151013836167225459784795}{425745373237801105805588685521690934871035473264662496426568} a^{2} + \frac{14793037730875743728796796075718610174814352214470929739021}{212872686618900552902794342760845467435517736632331248213284} a - \frac{7895981942933166016555956057759579740875925120989571317184}{53218171654725138225698585690211366858879434158082812053321}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4810}$, which has order $153920$ (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:  \( 205236.825908 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-455}) \), 3.3.169.1, 3.3.8281.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 6.0.15919187375.2, 6.0.780040181375.1, 6.0.780040181375.2, 6.0.4615622375.1, 9.9.567869252041.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$