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Properties

Label 18.8.94714551467...7968.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{12}\cdot 3^{6}\cdot 7^{13}\cdot 41^{9}$
Root discriminant $59.77$
Ramified primes $2, 3, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4493, -15253, -53273, -26905, -56371, -77665, -56755, -33496, 60144, 13996, -14475, 5201, -3254, 140, 346, 8, -20, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 20*x^16 + 8*x^15 + 346*x^14 + 140*x^13 - 3254*x^12 + 5201*x^11 - 14475*x^10 + 13996*x^9 + 60144*x^8 - 33496*x^7 - 56755*x^6 - 77665*x^5 - 56371*x^4 - 26905*x^3 - 53273*x^2 - 15253*x + 4493)
 
gp: K = bnfinit(x^18 - 3*x^17 - 20*x^16 + 8*x^15 + 346*x^14 + 140*x^13 - 3254*x^12 + 5201*x^11 - 14475*x^10 + 13996*x^9 + 60144*x^8 - 33496*x^7 - 56755*x^6 - 77665*x^5 - 56371*x^4 - 26905*x^3 - 53273*x^2 - 15253*x + 4493, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 20 x^{16} + 8 x^{15} + 346 x^{14} + 140 x^{13} - 3254 x^{12} + 5201 x^{11} - 14475 x^{10} + 13996 x^{9} + 60144 x^{8} - 33496 x^{7} - 56755 x^{6} - 77665 x^{5} - 56371 x^{4} - 26905 x^{3} - 53273 x^{2} - 15253 x + 4493 \)

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:  \(-94714551467214614836461979987968=-\,2^{12}\cdot 3^{6}\cdot 7^{13}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.77$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{2}{7} a^{14} - \frac{2}{7} a^{13} + \frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{77} a^{16} + \frac{4}{77} a^{15} + \frac{37}{77} a^{14} - \frac{24}{77} a^{13} - \frac{25}{77} a^{12} + \frac{6}{77} a^{11} - \frac{32}{77} a^{10} - \frac{30}{77} a^{9} + \frac{1}{11} a^{8} - \frac{36}{77} a^{7} + \frac{13}{77} a^{6} + \frac{2}{11} a^{5} + \frac{18}{77} a^{4} + \frac{3}{11} a^{3} - \frac{31}{77} a^{2} - \frac{36}{77} a + \frac{37}{77}$, $\frac{1}{4295421403994131878029567864244617802635527} a^{17} - \frac{2815395231801378528785050047291221014236}{4295421403994131878029567864244617802635527} a^{16} - \frac{135587970876155603336408809766573859710061}{4295421403994131878029567864244617802635527} a^{15} + \frac{126668709334503572169756784196187367179144}{4295421403994131878029567864244617802635527} a^{14} + \frac{230467499701269274945334571591317731113081}{4295421403994131878029567864244617802635527} a^{13} + \frac{24271708870402217061983519032780656963649}{55784693558365349065319063172008023410851} a^{12} - \frac{20822256911646018573555413992262671732424}{613631629142018839718509694892088257519361} a^{11} - \frac{254905950996026280257297205733321759249320}{613631629142018839718509694892088257519361} a^{10} + \frac{1165674326891068044399110656014838636911049}{4295421403994131878029567864244617802635527} a^{9} + \frac{155488511774942082724951528585292489358737}{390492854908557443457233442204056163875957} a^{8} - \frac{1543175955086994615019752805892248144643105}{4295421403994131878029567864244617802635527} a^{7} - \frac{240931308165709399631371719278430397152174}{613631629142018839718509694892088257519361} a^{6} + \frac{1955626198691211815930376295388981446425775}{4295421403994131878029567864244617802635527} a^{5} + \frac{1826870921440016660321171524509550805612206}{4295421403994131878029567864244617802635527} a^{4} - \frac{37224940617121770290890495684960169701926}{390492854908557443457233442204056163875957} a^{3} + \frac{866062659447057215684144122124452254700458}{4295421403994131878029567864244617802635527} a^{2} + \frac{107429673222669855715792801551312764827599}{4295421403994131878029567864244617802635527} a + \frac{172697059878760116932615775933598769582902}{613631629142018839718509694892088257519361}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 3587073135.27 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
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}$
41Data not computed