Properties

Label 18.0.45583107465...0263.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 229\cdot 433^{3}\cdot 12457$
Root discriminant $10.88$
Ramified primes $3, 229, 433, 12457$
Class number $1$
Class group Trivial
Galois Group 18T951

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, -1, 2, -2, 9, -10, 18, -20, 23, -20, 22, -10, 15, -2, 6, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 - 2*x^15 + 15*x^14 - 10*x^13 + 22*x^12 - 20*x^11 + 23*x^10 - 20*x^9 + 18*x^8 - 10*x^7 + 9*x^6 - 2*x^5 + 2*x^4 - x^3 - x + 1)
gp: K = bnfinit(x^18 + 6*x^16 - 2*x^15 + 15*x^14 - 10*x^13 + 22*x^12 - 20*x^11 + 23*x^10 - 20*x^9 + 18*x^8 - 10*x^7 + 9*x^6 - 2*x^5 + 2*x^4 - x^3 - x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut +\mathstrut 6 x^{16} \) \(\mathstrut -\mathstrut 2 x^{15} \) \(\mathstrut +\mathstrut 15 x^{14} \) \(\mathstrut -\mathstrut 10 x^{13} \) \(\mathstrut +\mathstrut 22 x^{12} \) \(\mathstrut -\mathstrut 20 x^{11} \) \(\mathstrut +\mathstrut 23 x^{10} \) \(\mathstrut -\mathstrut 20 x^{9} \) \(\mathstrut +\mathstrut 18 x^{8} \) \(\mathstrut -\mathstrut 10 x^{7} \) \(\mathstrut +\mathstrut 9 x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut +\mathstrut 2 x^{4} \) \(\mathstrut -\mathstrut x^{3} \) \(\mathstrut -\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-4558310746519570263=-\,3^{9}\cdot 229\cdot 433^{3}\cdot 12457\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.88$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 229, 433, 12457$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} - \frac{2}{13} a^{16} - \frac{3}{13} a^{15} + \frac{4}{13} a^{14} - \frac{6}{13} a^{13} + \frac{2}{13} a^{12} + \frac{5}{13} a^{11} - \frac{4}{13} a^{10} + \frac{5}{13} a^{9} - \frac{4}{13} a^{8} + \frac{3}{13} a^{6} + \frac{3}{13} a^{5} + \frac{5}{13} a^{4} + \frac{5}{13} a^{3} + \frac{2}{13} a^{2} - \frac{4}{13} a - \frac{6}{13}$

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:  \( a^{15} + 5 a^{13} - a^{12} + 10 a^{11} - 4 a^{10} + 11 a^{9} - 6 a^{8} + 8 a^{7} - 4 a^{6} + 4 a^{5} - a^{4} + 2 a^{3} + a \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a^{12} + 4 a^{10} - a^{9} + 6 a^{8} - 3 a^{7} + 5 a^{6} - 3 a^{5} + 3 a^{4} - a^{3} + a^{2} + 1 \),  \( a \),  \( \frac{6}{13} a^{17} + \frac{1}{13} a^{16} + \frac{34}{13} a^{15} - \frac{2}{13} a^{14} + \frac{81}{13} a^{13} - \frac{14}{13} a^{12} + \frac{108}{13} a^{11} - \frac{24}{13} a^{10} + \frac{82}{13} a^{9} - \frac{24}{13} a^{8} + 2 a^{7} - \frac{8}{13} a^{6} + \frac{5}{13} a^{5} + \frac{17}{13} a^{4} + \frac{17}{13} a^{3} - \frac{1}{13} a^{2} + \frac{2}{13} a - \frac{10}{13} \),  \( \frac{6}{13} a^{17} + \frac{1}{13} a^{16} + \frac{34}{13} a^{15} - \frac{2}{13} a^{14} + \frac{81}{13} a^{13} - \frac{27}{13} a^{12} + \frac{108}{13} a^{11} - \frac{63}{13} a^{10} + \frac{95}{13} a^{9} - \frac{50}{13} a^{8} + 4 a^{7} + \frac{5}{13} a^{6} + \frac{5}{13} a^{5} + \frac{30}{13} a^{4} - \frac{9}{13} a^{3} + \frac{12}{13} a^{2} + \frac{2}{13} a + \frac{3}{13} \),  \( \frac{2}{13} a^{17} - \frac{4}{13} a^{16} + \frac{20}{13} a^{15} - \frac{18}{13} a^{14} + \frac{79}{13} a^{13} - \frac{48}{13} a^{12} + \frac{153}{13} a^{11} - \frac{99}{13} a^{10} + \frac{166}{13} a^{9} - \frac{125}{13} a^{8} + 9 a^{7} - \frac{72}{13} a^{6} + \frac{71}{13} a^{5} - \frac{16}{13} a^{4} + \frac{36}{13} a^{3} - \frac{9}{13} a^{2} + \frac{18}{13} a - \frac{12}{13} \),  \( \frac{7}{13} a^{17} + \frac{12}{13} a^{16} + \frac{44}{13} a^{15} + \frac{54}{13} a^{14} + \frac{101}{13} a^{13} + \frac{92}{13} a^{12} + \frac{113}{13} a^{11} + \frac{76}{13} a^{10} + \frac{74}{13} a^{9} + \frac{37}{13} a^{8} + 4 a^{7} + \frac{21}{13} a^{6} + \frac{47}{13} a^{5} + \frac{22}{13} a^{4} + \frac{35}{13} a^{3} + \frac{14}{13} a^{2} + \frac{11}{13} a - \frac{3}{13} \),  \( a^{9} + 3 a^{7} - a^{6} + 3 a^{5} - 2 a^{4} + 2 a^{3} - a^{2} + a \),  \( \frac{1}{13} a^{17} - \frac{2}{13} a^{16} - \frac{3}{13} a^{15} - \frac{9}{13} a^{14} - \frac{32}{13} a^{13} + \frac{2}{13} a^{12} - \frac{86}{13} a^{11} + \frac{61}{13} a^{10} - \frac{125}{13} a^{9} + \frac{113}{13} a^{8} - 9 a^{7} + \frac{81}{13} a^{6} - \frac{62}{13} a^{5} + \frac{5}{13} a^{4} - \frac{8}{13} a^{3} - \frac{24}{13} a^{2} + \frac{9}{13} a + \frac{7}{13} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 147.731826961 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T951:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 3359232
The 275 conjugacy class representatives for t18n951 are not computed
Character table for t18n951 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 6.0.11691.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $18$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
229Data not computed
433Data not computed
12457Data not computed