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Properties

Label 18.10.1276091230...2421.3
Degree $18$
Signature $[10, 4]$
Discriminant $7^{12}\cdot 83^{4}\cdot 181^{5}$
Root discriminant $41.40$
Ramified primes $7, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T705

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7441, -3486, -4060, -9393, -1663, 1645, 2947, 4890, 1266, 370, -885, -495, -70, -105, 103, 45, -19, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 19*x^16 + 45*x^15 + 103*x^14 - 105*x^13 - 70*x^12 - 495*x^11 - 885*x^10 + 370*x^9 + 1266*x^8 + 4890*x^7 + 2947*x^6 + 1645*x^5 - 1663*x^4 - 9393*x^3 - 4060*x^2 - 3486*x + 7441)
 
gp: K = bnfinit(x^18 - 3*x^17 - 19*x^16 + 45*x^15 + 103*x^14 - 105*x^13 - 70*x^12 - 495*x^11 - 885*x^10 + 370*x^9 + 1266*x^8 + 4890*x^7 + 2947*x^6 + 1645*x^5 - 1663*x^4 - 9393*x^3 - 4060*x^2 - 3486*x + 7441, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 19 x^{16} + 45 x^{15} + 103 x^{14} - 105 x^{13} - 70 x^{12} - 495 x^{11} - 885 x^{10} + 370 x^{9} + 1266 x^{8} + 4890 x^{7} + 2947 x^{6} + 1645 x^{5} - 1663 x^{4} - 9393 x^{3} - 4060 x^{2} - 3486 x + 7441 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(127609123015304468465889942421=7^{12}\cdot 83^{4}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.40$
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:  $7, 83, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{2}{7} a^{14} - \frac{3}{7} a^{13} - \frac{3}{7} a^{12} + \frac{3}{7} a^{11} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3}$, $\frac{1}{210806872312237418775890833717489} a^{17} - \frac{8739117696960137942978424674078}{210806872312237418775890833717489} a^{16} - \frac{10216715568252988866890420881491}{210806872312237418775890833717489} a^{15} - \frac{5991086404096430185644880457850}{30115267473176774110841547673927} a^{14} - \frac{57721397824702947543918273777068}{210806872312237418775890833717489} a^{13} - \frac{91855075553122737901466307692620}{210806872312237418775890833717489} a^{12} + \frac{8292645629282450727919250644195}{210806872312237418775890833717489} a^{11} - \frac{91953288385776043251027986636635}{210806872312237418775890833717489} a^{10} + \frac{24562431961266936633539867601499}{210806872312237418775890833717489} a^{9} - \frac{902386858120028599039685748997}{4302181067596682015834506810561} a^{8} - \frac{86593922201194953764932127909173}{210806872312237418775890833717489} a^{7} + \frac{91626998491622813152588311111447}{210806872312237418775890833717489} a^{6} + \frac{50866842012654235391061175871117}{210806872312237418775890833717489} a^{5} + \frac{31536850544233222113929131819001}{210806872312237418775890833717489} a^{4} + \frac{50180244849126517067129557068080}{210806872312237418775890833717489} a^{3} - \frac{1023409668785105835906843944412}{30115267473176774110841547673927} a^{2} + \frac{114171523667245408520047661149}{2316559036398213393141657513379} a + \frac{13220827465457423859349514628139}{30115267473176774110841547673927}$

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:  $13$
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:  \( 201728420.142 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T705:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 64 conjugacy class representatives for t18n705 are not computed
Character table for t18n705 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $18$ $18$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.3.2.1$x^{3} - 83$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
83.3.2.1$x^{3} - 83$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
181Data not computed