↑

Properties

Label 18.12.1019443660...4128.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{16}\cdot 37^{9}\cdot 67^{2}\cdot 139\cdot 4999^{2}\cdot 87613^{2}$
Root discriminant $215.67$
Ramified primes $2, 37, 67, 139, 4999, 87613$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T874

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![698831764, -2812072848, 4580574604, -3689296500, 1202258516, 310262532, -390842556, 86724264, 26645874, -15086532, 491516, 949740, -135655, -28474, 6435, 396, -131, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 131*x^16 + 396*x^15 + 6435*x^14 - 28474*x^13 - 135655*x^12 + 949740*x^11 + 491516*x^10 - 15086532*x^9 + 26645874*x^8 + 86724264*x^7 - 390842556*x^6 + 310262532*x^5 + 1202258516*x^4 - 3689296500*x^3 + 4580574604*x^2 - 2812072848*x + 698831764)
 
gp: K = bnfinit(x^18 - 2*x^17 - 131*x^16 + 396*x^15 + 6435*x^14 - 28474*x^13 - 135655*x^12 + 949740*x^11 + 491516*x^10 - 15086532*x^9 + 26645874*x^8 + 86724264*x^7 - 390842556*x^6 + 310262532*x^5 + 1202258516*x^4 - 3689296500*x^3 + 4580574604*x^2 - 2812072848*x + 698831764, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 131 x^{16} + 396 x^{15} + 6435 x^{14} - 28474 x^{13} - 135655 x^{12} + 949740 x^{11} + 491516 x^{10} - 15086532 x^{9} + 26645874 x^{8} + 86724264 x^{7} - 390842556 x^{6} + 310262532 x^{5} + 1202258516 x^{4} - 3689296500 x^{3} + 4580574604 x^{2} - 2812072848 x + 698831764 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1019443660108235620242452440038509695664128=-\,2^{16}\cdot 37^{9}\cdot 67^{2}\cdot 139\cdot 4999^{2}\cdot 87613^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $215.67$
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:  $2, 37, 67, 139, 4999, 87613$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{16} a^{15} + \frac{1}{16} a^{14} + \frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{32} a^{16} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{32} a^{12} - \frac{1}{4} a^{11} - \frac{3}{16} a^{10} + \frac{1}{16} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{2696775220554594370611458412672488707284334720} a^{17} - \frac{7093095331667314099486564972894327327652869}{539355044110918874122291682534497741456866944} a^{16} - \frac{5625322650348398124733175817096466586219009}{674193805138648592652864603168122176821083680} a^{15} + \frac{34581517347229036887663039588567389361077703}{337096902569324296326432301584061088410541840} a^{14} - \frac{325862943880318858546803867596114930302412757}{2696775220554594370611458412672488707284334720} a^{13} + \frac{164661047909057995928163484091344071516231577}{2696775220554594370611458412672488707284334720} a^{12} + \frac{126473613109405744469372210840480066327316357}{1348387610277297185305729206336244353642167360} a^{11} - \frac{209177554857452339018410458807581646158311181}{1348387610277297185305729206336244353642167360} a^{10} - \frac{139866121046148540775070281693434277521534119}{1348387610277297185305729206336244353642167360} a^{9} + \frac{29871931921359485797368884451522285363002831}{1348387610277297185305729206336244353642167360} a^{8} - \frac{2307967206313524614039509041438298377879461}{84274225642331074081608075396015272102635460} a^{7} - \frac{7300437276604305533793326575073917662640307}{67419380513864859265286460316812217682108368} a^{6} - \frac{65452545645452597311674279919014120888489549}{674193805138648592652864603168122176821083680} a^{5} + \frac{16392526653837230148464712579832584793943939}{33709690256932429632643230158406108841054184} a^{4} - \frac{203592176381379999595208991533438267638329711}{674193805138648592652864603168122176821083680} a^{3} - \frac{70401214331323637204361741233799263511909473}{168548451284662148163216150792030544205270920} a^{2} + \frac{199181244006546242432921766489108833368450767}{674193805138648592652864603168122176821083680} a - \frac{203579325427818825947413170226993997276815053}{674193805138648592652864603168122176821083680}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $14$
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:  \( 251772484727000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T874:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 174 conjugacy class representatives for t18n874 are not computed
Character table for t18n874 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 6.6.810448.1

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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.12.27$x^{12} - 18 x^{10} + 171 x^{8} + 116 x^{6} - 313 x^{4} + 190 x^{2} + 877$$6$$2$$12$12T30$[4/3, 4/3]_{3}^{4}$
37Data not computed
67Data not computed
139Data not computed
4999Data not computed
87613Data not computed