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Properties

Label 18.10.2799928238...217.10
Degree $18$
Signature $[10, 4]$
Discriminant $7^{12}\cdot 53^{6}\cdot 97^{3}$
Root discriminant $29.46$
Ramified primes $7, 53, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T544

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-97, -339, 4187, -10550, 11175, -4370, -4662, 9387, -7706, 3569, 102, -1176, 360, 38, 13, -19, 9, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 9*x^16 - 19*x^15 + 13*x^14 + 38*x^13 + 360*x^12 - 1176*x^11 + 102*x^10 + 3569*x^9 - 7706*x^8 + 9387*x^7 - 4662*x^6 - 4370*x^5 + 11175*x^4 - 10550*x^3 + 4187*x^2 - 339*x - 97)
 
gp: K = bnfinit(x^18 - 5*x^17 + 9*x^16 - 19*x^15 + 13*x^14 + 38*x^13 + 360*x^12 - 1176*x^11 + 102*x^10 + 3569*x^9 - 7706*x^8 + 9387*x^7 - 4662*x^6 - 4370*x^5 + 11175*x^4 - 10550*x^3 + 4187*x^2 - 339*x - 97, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 9 x^{16} - 19 x^{15} + 13 x^{14} + 38 x^{13} + 360 x^{12} - 1176 x^{11} + 102 x^{10} + 3569 x^{9} - 7706 x^{8} + 9387 x^{7} - 4662 x^{6} - 4370 x^{5} + 11175 x^{4} - 10550 x^{3} + 4187 x^{2} - 339 x - 97 \)

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:  \(279992823820843547402730217=7^{12}\cdot 53^{6}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.46$
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:  $7, 53, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{44} a^{15} - \frac{7}{22} a^{14} - \frac{1}{4} a^{13} + \frac{21}{44} a^{12} - \frac{4}{11} a^{11} + \frac{9}{44} a^{10} - \frac{3}{11} a^{9} - \frac{9}{22} a^{8} - \frac{15}{44} a^{7} + \frac{5}{11} a^{6} - \frac{3}{22} a^{5} + \frac{3}{22} a^{4} - \frac{2}{11} a^{3} + \frac{2}{11} a^{2} + \frac{5}{44} a + \frac{1}{44}$, $\frac{1}{44} a^{16} + \frac{13}{44} a^{14} - \frac{1}{44} a^{13} + \frac{7}{22} a^{12} + \frac{5}{44} a^{11} - \frac{9}{22} a^{10} - \frac{5}{22} a^{9} - \frac{3}{44} a^{8} - \frac{7}{22} a^{7} + \frac{5}{22} a^{6} + \frac{5}{22} a^{5} - \frac{3}{11} a^{4} - \frac{4}{11} a^{3} - \frac{15}{44} a^{2} - \frac{17}{44} a + \frac{7}{22}$, $\frac{1}{1142843206418854237819810843588} a^{17} + \frac{12464968548660217874194990995}{1142843206418854237819810843588} a^{16} - \frac{4794007142862049160454570695}{1142843206418854237819810843588} a^{15} - \frac{265054017292246775964232370367}{571421603209427118909905421794} a^{14} - \frac{10925325445303630123029077711}{103894836947168567074528258508} a^{13} + \frac{514822756658551134802723279111}{1142843206418854237819810843588} a^{12} - \frac{10526754625197301660299206229}{103894836947168567074528258508} a^{11} - \frac{116224008856479845422890833789}{285710801604713559454952710897} a^{10} + \frac{459683924199637214019971990483}{1142843206418854237819810843588} a^{9} - \frac{403841340239737681830370472683}{1142843206418854237819810843588} a^{8} + \frac{51836779059770276353831539685}{285710801604713559454952710897} a^{7} - \frac{94810865645855795562636207284}{285710801604713559454952710897} a^{6} - \frac{3787872513319480962710135547}{571421603209427118909905421794} a^{5} - \frac{69103990739514886581584398189}{285710801604713559454952710897} a^{4} + \frac{11972317085722742744370500377}{1142843206418854237819810843588} a^{3} + \frac{70220347449330020077786162109}{571421603209427118909905421794} a^{2} - \frac{263656609017555088796616917153}{1142843206418854237819810843588} a - \frac{2394872304325194534961951249}{5890944362983784731029952802}$

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:  \( 4433161.15377 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T544:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 9216
The 96 conjugacy class representatives for t18n544 are not computed
Character table for t18n544 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
7Data not computed
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
97Data not computed