Properties

Label 18.0.51327388829...6176.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{12}\cdot 11^{9}$
Root discriminant $10.95$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois Group $C_3\times C_3:S_3$ (as 18T23)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 29, -79, 154, -224, 253, -232, 181, -108, 16, 64, -87, 51, -2, -20, 16, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 16*x^16 - 20*x^15 - 2*x^14 + 51*x^13 - 87*x^12 + 64*x^11 + 16*x^10 - 108*x^9 + 181*x^8 - 232*x^7 + 253*x^6 - 224*x^5 + 154*x^4 - 79*x^3 + 29*x^2 - 7*x + 1)
gp: K = bnfinit(x^18 - 6*x^17 + 16*x^16 - 20*x^15 - 2*x^14 + 51*x^13 - 87*x^12 + 64*x^11 + 16*x^10 - 108*x^9 + 181*x^8 - 232*x^7 + 253*x^6 - 224*x^5 + 154*x^4 - 79*x^3 + 29*x^2 - 7*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 6 x^{17} \) \(\mathstrut +\mathstrut 16 x^{16} \) \(\mathstrut -\mathstrut 20 x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut 51 x^{13} \) \(\mathstrut -\mathstrut 87 x^{12} \) \(\mathstrut +\mathstrut 64 x^{11} \) \(\mathstrut +\mathstrut 16 x^{10} \) \(\mathstrut -\mathstrut 108 x^{9} \) \(\mathstrut +\mathstrut 181 x^{8} \) \(\mathstrut -\mathstrut 232 x^{7} \) \(\mathstrut +\mathstrut 253 x^{6} \) \(\mathstrut -\mathstrut 224 x^{5} \) \(\mathstrut +\mathstrut 154 x^{4} \) \(\mathstrut -\mathstrut 79 x^{3} \) \(\mathstrut +\mathstrut 29 x^{2} \) \(\mathstrut -\mathstrut 7 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-5132738882980786176=-\,2^{12}\cdot 3^{12}\cdot 11^{9}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.95$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{2733} a^{17} + \frac{233}{2733} a^{16} + \frac{44}{911} a^{15} - \frac{119}{911} a^{14} + \frac{103}{911} a^{13} + \frac{37}{911} a^{12} + \frac{934}{2733} a^{11} - \frac{272}{911} a^{10} + \frac{857}{2733} a^{9} - \frac{1171}{2733} a^{8} + \frac{300}{911} a^{7} - \frac{128}{2733} a^{6} - \frac{1187}{2733} a^{5} + \frac{1226}{2733} a^{4} - \frac{58}{911} a^{3} - \frac{670}{2733} a^{2} - \frac{676}{2733} a + \frac{587}{2733}$

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

Class group and class number

Trivial Abelian group, 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:  \( \frac{5708}{2733} a^{17} - \frac{28337}{2733} a^{16} + \frac{57452}{2733} a^{15} - \frac{29912}{2733} a^{14} - \frac{33378}{911} a^{13} + \frac{239125}{2733} a^{12} - \frac{193943}{2733} a^{11} - \frac{17541}{911} a^{10} + \frac{98587}{911} a^{9} - \frac{392702}{2733} a^{8} + \frac{143658}{911} a^{7} - \frac{470989}{2733} a^{6} + \frac{138983}{911} a^{5} - \frac{211646}{2733} a^{4} + \frac{7828}{911} a^{3} + \frac{48301}{2733} a^{2} - \frac{32408}{2733} a + \frac{3016}{911} \),  \( \frac{1423}{911} a^{17} - \frac{23821}{2733} a^{16} + \frac{56081}{2733} a^{15} - \frac{16982}{911} a^{14} - \frac{17615}{911} a^{13} + \frac{70497}{911} a^{12} - \frac{253459}{2733} a^{11} + \frac{26776}{911} a^{10} + \frac{191267}{2733} a^{9} - \frac{397538}{2733} a^{8} + \frac{174746}{911} a^{7} - \frac{615668}{2733} a^{6} + \frac{204868}{911} a^{5} - \frac{455401}{2733} a^{4} + \frac{233786}{2733} a^{3} - \frac{74392}{2733} a^{2} + \frac{6581}{2733} a + \frac{1564}{2733} \),  \( \frac{4097}{2733} a^{17} - \frac{7634}{911} a^{16} + \frac{56152}{2733} a^{15} - \frac{56956}{2733} a^{14} - \frac{41312}{2733} a^{13} + \frac{70510}{911} a^{12} - \frac{93093}{911} a^{11} + \frac{113182}{2733} a^{10} + \frac{190534}{2733} a^{9} - \frac{424787}{2733} a^{8} + \frac{548905}{2733} a^{7} - \frac{636469}{2733} a^{6} + \frac{655699}{2733} a^{5} - \frac{503204}{2733} a^{4} + \frac{253693}{2733} a^{3} - \frac{66650}{2733} a^{2} + \frac{6245}{2733} a + \frac{1721}{2733} \),  \( \frac{383}{911} a^{17} - \frac{1028}{2733} a^{16} - \frac{11401}{2733} a^{15} + \frac{41663}{2733} a^{14} - \frac{51265}{2733} a^{13} - \frac{5770}{911} a^{12} + \frac{47982}{911} a^{11} - \frac{190564}{2733} a^{10} + \frac{79159}{2733} a^{9} + \frac{104833}{2733} a^{8} - \frac{235834}{2733} a^{7} + \frac{321182}{2733} a^{6} - \frac{398203}{2733} a^{5} + \frac{138865}{911} a^{4} - \frac{308335}{2733} a^{3} + \frac{157568}{2733} a^{2} - \frac{46102}{2733} a + \frac{2537}{911} \),  \( \frac{4192}{2733} a^{17} - \frac{6329}{911} a^{16} + \frac{37718}{2733} a^{15} - \frac{7819}{911} a^{14} - \frac{16436}{911} a^{13} + \frac{139174}{2733} a^{12} - \frac{144088}{2733} a^{11} + \frac{35662}{2733} a^{10} + \frac{128011}{2733} a^{9} - \frac{261821}{2733} a^{8} + \frac{357461}{2733} a^{7} - \frac{408125}{2733} a^{6} + \frac{394441}{2733} a^{5} - \frac{96419}{911} a^{4} + \frac{53850}{911} a^{3} - \frac{16407}{911} a^{2} + \frac{9439}{2733} a + \frac{1915}{2733} \),  \( a \),  \( \frac{2608}{2733} a^{17} - \frac{11816}{2733} a^{16} + \frac{7254}{911} a^{15} - \frac{2434}{911} a^{14} - \frac{43180}{2733} a^{13} + \frac{89068}{2733} a^{12} - \frac{63001}{2733} a^{11} - \frac{27362}{2733} a^{10} + \frac{36868}{911} a^{9} - \frac{145145}{2733} a^{8} + \frac{54511}{911} a^{7} - \frac{176221}{2733} a^{6} + \frac{151108}{2733} a^{5} - \frac{24968}{911} a^{4} + \frac{9907}{2733} a^{3} + \frac{5749}{911} a^{2} - \frac{10244}{2733} a + \frac{1327}{2733} \),  \( \frac{6830}{2733} a^{17} - \frac{12189}{911} a^{16} + \frac{83482}{2733} a^{15} - \frac{70621}{2733} a^{14} - \frac{87773}{2733} a^{13} + \frac{106039}{911} a^{12} - \frac{121334}{911} a^{11} + \frac{96784}{2733} a^{10} + \frac{302587}{2733} a^{9} - \frac{591500}{2733} a^{8} + \frac{764812}{2733} a^{7} - \frac{887905}{2733} a^{6} + \frac{879805}{2733} a^{5} - \frac{639854}{2733} a^{4} + \frac{313819}{2733} a^{3} - \frac{85781}{2733} a^{2} + \frac{6245}{2733} a + \frac{4454}{2733} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 51.3669704577 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.44.1 x3, 6.0.107811.1, 6.0.1724976.2, 6.0.1724976.1, 6.0.21296.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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

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$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$