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Properties

Label 18.2.402...137.1
Degree $18$
Signature $[2, 8]$
Discriminant $4.025\times 10^{40}$
Root discriminant $180.23$
Ramified prime $17$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\PGL(2,17)$ (as 18T468)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 17*x^16 + 17*x^15 - 935*x^14 + 799*x^13 + 9231*x^12 - 41463*x^11 + 192780*x^10 + 291686*x^9 - 390014*x^8 + 6132223*x^7 - 3955645*x^6 + 2916112*x^5 + 45030739*x^4 - 94452714*x^3 + 184016925*x^2 - 141466230*x + 113422599)
 
gp: K = bnfinit(x^18 - 7*x^17 + 17*x^16 + 17*x^15 - 935*x^14 + 799*x^13 + 9231*x^12 - 41463*x^11 + 192780*x^10 + 291686*x^9 - 390014*x^8 + 6132223*x^7 - 3955645*x^6 + 2916112*x^5 + 45030739*x^4 - 94452714*x^3 + 184016925*x^2 - 141466230*x + 113422599, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![113422599, -141466230, 184016925, -94452714, 45030739, 2916112, -3955645, 6132223, -390014, 291686, 192780, -41463, 9231, 799, -935, 17, 17, -7, 1]);
 

\( x^{18} - 7 x^{17} + 17 x^{16} + 17 x^{15} - 935 x^{14} + 799 x^{13} + 9231 x^{12} - 41463 x^{11} + 192780 x^{10} + 291686 x^{9} - 390014 x^{8} + 6132223 x^{7} - 3955645 x^{6} + 2916112 x^{5} + 45030739 x^{4} - 94452714 x^{3} + 184016925 x^{2} - 141466230 x + 113422599 \)

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(40254497110927943179349807054456171205137\)\(\medspace = 17^{33}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $180.23$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} - \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{11} - \frac{1}{27} a^{10} + \frac{1}{27} a^{9} + \frac{1}{27} a^{8} - \frac{1}{27} a^{7} - \frac{1}{27} a^{6} + \frac{4}{27} a^{5} - \frac{1}{3} a^{4} - \frac{1}{27} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{14} - \frac{1}{27} a^{12} - \frac{1}{27} a^{11} + \frac{1}{27} a^{10} + \frac{1}{27} a^{9} - \frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{4}{27} a^{6} - \frac{10}{27} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{15} - \frac{1}{27} a^{12} + \frac{1}{9} a^{7} - \frac{1}{27} a^{6} + \frac{1}{9} a^{5} - \frac{2}{9} a^{4} - \frac{1}{27} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{11} - \frac{1}{27} a^{10} + \frac{1}{27} a^{9} + \frac{4}{27} a^{8} - \frac{2}{27} a^{7} + \frac{2}{27} a^{6} - \frac{2}{27} a^{5} - \frac{10}{27} a^{4} - \frac{7}{27} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{7094103080282041057010466744591157150275076597906943206625637} a^{17} + \frac{33306058218965096825406300754062419755021319837762746981781}{7094103080282041057010466744591157150275076597906943206625637} a^{16} + \frac{113280437735221180596099517187054393793903590588158249434011}{7094103080282041057010466744591157150275076597906943206625637} a^{15} + \frac{63477218186831708295339371125762892490906614152752133861748}{7094103080282041057010466744591157150275076597906943206625637} a^{14} - \frac{74373966962318072835903796536680710457213920606273283465810}{7094103080282041057010466744591157150275076597906943206625637} a^{13} + \frac{26422156635380679168571928836057583643726777235204490215900}{7094103080282041057010466744591157150275076597906943206625637} a^{12} + \frac{1011544388066388771488958382490728422938368390356278837669}{87581519509654827864326749933224162349075019727246212427477} a^{11} - \frac{12500322977847165381105112558806468502702489228971942713135}{788233675586893450778940749399017461141675177545215911847293} a^{10} + \frac{53918180854577885340229395640194144607694034706814562804904}{788233675586893450778940749399017461141675177545215911847293} a^{9} + \frac{485286017716920693113743662888986979959906195893717959196193}{7094103080282041057010466744591157150275076597906943206625637} a^{8} + \frac{573632909541730501101989271238884464535064874448775160698546}{7094103080282041057010466744591157150275076597906943206625637} a^{7} + \frac{18559627702323962639235972978530333324815977275549071584223}{7094103080282041057010466744591157150275076597906943206625637} a^{6} + \frac{605095082370349487028612855302968143662809853644242341385755}{7094103080282041057010466744591157150275076597906943206625637} a^{5} - \frac{3492989685279513546436257846936644802781664944539705228262751}{7094103080282041057010466744591157150275076597906943206625637} a^{4} - \frac{50718107243063419281946547030748626781197034471958243437374}{7094103080282041057010466744591157150275076597906943206625637} a^{3} - \frac{393375175373788355630462288850253498835816268184014763925975}{2364701026760680352336822248197052383425025532635647735541879} a^{2} + \frac{53813386405165044217750392174977292140937048316015113206813}{262744558528964483592980249799672487047225059181738637282431} a + \frac{1811201964882249481207172638891755105738601597851513211357}{262744558528964483592980249799672487047225059181738637282431}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 45413380405400 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{8}\cdot 45413380405400 \cdot 1}{2\sqrt{40254497110927943179349807054456171205137}}\approx 1.09962744387092$ (assuming GRH)

Galois group

$\PGL(2,17)$ (as 18T468):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 4896
The 19 conjugacy class representatives for $\PGL(2,17)$
Character table for $\PGL(2,17)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ $16{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ $18$ $16{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
17Data not computed

Additional information

This field is associated with the 17-division points on any elliptic curve in the isogeny class 17.a1.