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Properties

Label 18.0.41040300635...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{9}\cdot 7^{12}\cdot 19^{15}$
Root discriminant $95.17$
Ramified primes $5, 7, 19$
Class number $37856$ (GRH)
Class group $[2, 26, 728]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14492736, -16477776, 29682552, -27033887, 21097703, -12462682, 6247014, -1953586, 647468, -407056, 256538, -86808, 25794, -7266, 1402, -252, 52, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 52*x^16 - 252*x^15 + 1402*x^14 - 7266*x^13 + 25794*x^12 - 86808*x^11 + 256538*x^10 - 407056*x^9 + 647468*x^8 - 1953586*x^7 + 6247014*x^6 - 12462682*x^5 + 21097703*x^4 - 27033887*x^3 + 29682552*x^2 - 16477776*x + 14492736)
 
gp: K = bnfinit(x^18 - 3*x^17 + 52*x^16 - 252*x^15 + 1402*x^14 - 7266*x^13 + 25794*x^12 - 86808*x^11 + 256538*x^10 - 407056*x^9 + 647468*x^8 - 1953586*x^7 + 6247014*x^6 - 12462682*x^5 + 21097703*x^4 - 27033887*x^3 + 29682552*x^2 - 16477776*x + 14492736, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 52 x^{16} - 252 x^{15} + 1402 x^{14} - 7266 x^{13} + 25794 x^{12} - 86808 x^{11} + 256538 x^{10} - 407056 x^{9} + 647468 x^{8} - 1953586 x^{7} + 6247014 x^{6} - 12462682 x^{5} + 21097703 x^{4} - 27033887 x^{3} + 29682552 x^{2} - 16477776 x + 14492736 \)

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:  \(-410403006358127324860166545115234375=-\,5^{9}\cdot 7^{12}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.17$
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:  $5, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(665=5\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{665}(1,·)$, $\chi_{665}(449,·)$, $\chi_{665}(11,·)$, $\chi_{665}(464,·)$, $\chi_{665}(274,·)$, $\chi_{665}(596,·)$, $\chi_{665}(569,·)$, $\chi_{665}(284,·)$, $\chi_{665}(354,·)$, $\chi_{665}(571,·)$, $\chi_{665}(296,·)$, $\chi_{665}(106,·)$, $\chi_{665}(191,·)$, $\chi_{665}(179,·)$, $\chi_{665}(501,·)$, $\chi_{665}(121,·)$, $\chi_{665}(379,·)$, $\chi_{665}(639,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{22} a^{12} + \frac{1}{11} a^{11} + \frac{1}{11} a^{10} + \frac{5}{22} a^{9} - \frac{5}{22} a^{8} + \frac{3}{11} a^{7} - \frac{3}{11} a^{6} - \frac{3}{22} a^{5} + \frac{3}{11} a^{4} - \frac{1}{11} a^{3} - \frac{1}{22} a^{2} - \frac{1}{22} a - \frac{3}{11}$, $\frac{1}{22} a^{13} - \frac{1}{11} a^{11} + \frac{1}{22} a^{10} - \frac{2}{11} a^{9} + \frac{5}{22} a^{8} + \frac{2}{11} a^{7} + \frac{9}{22} a^{6} - \frac{5}{11} a^{5} + \frac{4}{11} a^{4} + \frac{3}{22} a^{3} - \frac{5}{11} a^{2} + \frac{7}{22} a - \frac{5}{11}$, $\frac{1}{44} a^{14} - \frac{1}{44} a^{13} - \frac{1}{44} a^{12} + \frac{5}{44} a^{11} - \frac{3}{44} a^{10} + \frac{3}{44} a^{9} + \frac{5}{44} a^{8} - \frac{1}{4} a^{7} + \frac{19}{44} a^{6} + \frac{15}{44} a^{5} + \frac{1}{44} a^{4} - \frac{15}{44} a^{3} - \frac{17}{44} a^{2} - \frac{7}{44} a + \frac{1}{11}$, $\frac{1}{8404} a^{15} + \frac{9}{8404} a^{14} - \frac{13}{8404} a^{13} - \frac{135}{8404} a^{12} + \frac{15}{764} a^{11} + \frac{1537}{8404} a^{10} - \frac{629}{8404} a^{9} - \frac{1389}{8404} a^{8} + \frac{3785}{8404} a^{7} - \frac{2641}{8404} a^{6} - \frac{357}{764} a^{5} - \frac{1175}{8404} a^{4} + \frac{3849}{8404} a^{3} + \frac{2151}{8404} a^{2} - \frac{745}{4202} a - \frac{703}{2101}$, $\frac{1}{100848} a^{16} - \frac{119}{25212} a^{14} + \frac{47}{4202} a^{13} + \frac{881}{50424} a^{12} - \frac{2033}{8404} a^{11} + \frac{9}{16808} a^{10} - \frac{1389}{16808} a^{9} - \frac{1489}{6303} a^{8} + \frac{4723}{12606} a^{7} - \frac{7741}{25212} a^{6} - \frac{2207}{50424} a^{5} + \frac{629}{8404} a^{4} + \frac{2473}{50424} a^{3} + \frac{11621}{100848} a^{2} - \frac{2513}{6303} a - \frac{189}{2101}$, $\frac{1}{122603328004871283664906736020391078061492737247700272} a^{17} + \frac{7406118114141296049258211030006029575973858495}{6811296000270626870272596445577282114527374291538904} a^{16} - \frac{159592058145728742129937534164778212092983907021}{7662708000304455229056671001274442378843296077981267} a^{15} + \frac{16383425456773948015706955012478505949419850989291}{5108472000202970152704447334182961585895530718654178} a^{14} + \frac{58732640579824455532614167253989727089905775785505}{61301664002435641832453368010195539030746368623850136} a^{13} + \frac{8264261026667117598339019400243061372941723298621}{851412000033828358784074555697160264315921786442363} a^{12} - \frac{883601398153368151505842164261491742401405263103705}{6811296000270626870272596445577282114527374291538904} a^{11} - \frac{2025738218228328263178804929064864330971708881928039}{20433888000811880610817789336731846343582122874616712} a^{10} + \frac{1582978106393155263304419784030621026263176311949259}{30650832001217820916226684005097769515373184311925068} a^{9} - \frac{3482054987382692274275550105184709606104106266908011}{15325416000608910458113342002548884757686592155962534} a^{8} + \frac{8031248429191573333720604497550913851828379412990145}{30650832001217820916226684005097769515373184311925068} a^{7} + \frac{5707598820819411552220955956509582617572737318532357}{61301664002435641832453368010195539030746368623850136} a^{6} - \frac{204696949527378340277049147507512070388101945083881}{5108472000202970152704447334182961585895530718654178} a^{5} - \frac{8082591389665076340788265081087200512410122770289123}{61301664002435641832453368010195539030746368623850136} a^{4} - \frac{47248925382627940657144812556344676165211801121887071}{122603328004871283664906736020391078061492737247700272} a^{3} + \frac{28314173943052309289707474538264237413664805645730139}{61301664002435641832453368010195539030746368623850136} a^{2} - \frac{413770777304201490409786985890239345325962018700943}{928813090945994573218990424396902106526460130664396} a + \frac{34785263298819950827427895838279404104775512243389}{851412000033828358784074555697160264315921786442363}$

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

Class group and class number

$C_{2}\times C_{26}\times C_{728}$, which has order $37856$ (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:  \( 833965.243856 \) (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

\(\Q(\sqrt{-95}) \), 3.3.17689.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 3.3.361.1, 6.0.743139212375.1, 6.0.743139212375.2, 6.0.2058557375.2, 6.0.309512375.1, 9.9.5534900853769.1

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$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}$
19Data not computed