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Properties

Label 18.0.15928224673...2976.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{6}\cdot 7^{12}\cdot 97^{2}$
Root discriminant $22.11$
Ramified primes $2, 3, 7, 97$
Class number $1$
Class group Trivial
Galois group 18T366

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4271, -17494, 37263, -48268, 42769, -24970, 9116, 116, -1924, 1818, -879, 570, -211, 86, 13, -18, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 14*x^16 - 18*x^15 + 13*x^14 + 86*x^13 - 211*x^12 + 570*x^11 - 879*x^10 + 1818*x^9 - 1924*x^8 + 116*x^7 + 9116*x^6 - 24970*x^5 + 42769*x^4 - 48268*x^3 + 37263*x^2 - 17494*x + 4271)
 
gp: K = bnfinit(x^18 - 4*x^17 + 14*x^16 - 18*x^15 + 13*x^14 + 86*x^13 - 211*x^12 + 570*x^11 - 879*x^10 + 1818*x^9 - 1924*x^8 + 116*x^7 + 9116*x^6 - 24970*x^5 + 42769*x^4 - 48268*x^3 + 37263*x^2 - 17494*x + 4271, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 14 x^{16} - 18 x^{15} + 13 x^{14} + 86 x^{13} - 211 x^{12} + 570 x^{11} - 879 x^{10} + 1818 x^{9} - 1924 x^{8} + 116 x^{7} + 9116 x^{6} - 24970 x^{5} + 42769 x^{4} - 48268 x^{3} + 37263 x^{2} - 17494 x + 4271 \)

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:  \(-1592822467387587324542976=-\,2^{24}\cdot 3^{6}\cdot 7^{12}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.11$
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:  $2, 3, 7, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{257601592263394719141459416401} a^{17} + \frac{73139212230452766173068069979}{257601592263394719141459416401} a^{16} + \frac{62816752344544948685225477152}{257601592263394719141459416401} a^{15} + \frac{27885799573341495613502895626}{257601592263394719141459416401} a^{14} - \frac{115299347767845271299615015925}{257601592263394719141459416401} a^{13} - \frac{91970635287149405203326321196}{257601592263394719141459416401} a^{12} + \frac{91723448787558284170766218629}{257601592263394719141459416401} a^{11} - \frac{63498675750997253835544504454}{257601592263394719141459416401} a^{10} + \frac{64091601899423684396058996822}{257601592263394719141459416401} a^{9} - \frac{64052872479626234659842994899}{257601592263394719141459416401} a^{8} + \frac{121338230397010727138654356155}{257601592263394719141459416401} a^{7} + \frac{74589015128616986304962910889}{257601592263394719141459416401} a^{6} + \frac{20869085274310808737415720976}{257601592263394719141459416401} a^{5} + \frac{12203680360100294757978180690}{257601592263394719141459416401} a^{4} - \frac{43130352058780737998086564777}{257601592263394719141459416401} a^{3} - \frac{76497231567763576973325179472}{257601592263394719141459416401} a^{2} + \frac{88633408850090013900667279405}{257601592263394719141459416401} a + \frac{127206120267046390685293296797}{257601592263394719141459416401}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 33357.0470844 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T366:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n366
Character table for t18n366 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.1.588.1, 9.3.203297472.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed
97Data not computed