Properties

Label 18.0.66464365673...9399.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,17^{2}\cdot 43^{2}\cdot 2311\cdot 73363^{2}$
Root discriminant $11.11$
Ramified primes $17, 43, 2311, 73363$
Class number $1$
Class group Trivial
Galois Group 18T968

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 14, -37, 85, -174, 321, -532, 787, -1020, 1139, -1082, 865, -574, 310, -132, 42, -9, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 42*x^16 - 132*x^15 + 310*x^14 - 574*x^13 + 865*x^12 - 1082*x^11 + 1139*x^10 - 1020*x^9 + 787*x^8 - 532*x^7 + 321*x^6 - 174*x^5 + 85*x^4 - 37*x^3 + 14*x^2 - 4*x + 1)
gp: K = bnfinit(x^18 - 9*x^17 + 42*x^16 - 132*x^15 + 310*x^14 - 574*x^13 + 865*x^12 - 1082*x^11 + 1139*x^10 - 1020*x^9 + 787*x^8 - 532*x^7 + 321*x^6 - 174*x^5 + 85*x^4 - 37*x^3 + 14*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 9 x^{17} \) \(\mathstrut +\mathstrut 42 x^{16} \) \(\mathstrut -\mathstrut 132 x^{15} \) \(\mathstrut +\mathstrut 310 x^{14} \) \(\mathstrut -\mathstrut 574 x^{13} \) \(\mathstrut +\mathstrut 865 x^{12} \) \(\mathstrut -\mathstrut 1082 x^{11} \) \(\mathstrut +\mathstrut 1139 x^{10} \) \(\mathstrut -\mathstrut 1020 x^{9} \) \(\mathstrut +\mathstrut 787 x^{8} \) \(\mathstrut -\mathstrut 532 x^{7} \) \(\mathstrut +\mathstrut 321 x^{6} \) \(\mathstrut -\mathstrut 174 x^{5} \) \(\mathstrut +\mathstrut 85 x^{4} \) \(\mathstrut -\mathstrut 37 x^{3} \) \(\mathstrut +\mathstrut 14 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-6646436567333419399=-\,17^{2}\cdot 43^{2}\cdot 2311\cdot 73363^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.11$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $17, 43, 2311, 73363$
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 \(a\))

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

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:  \( a^{2} - a + 1 \),  \( 2 a^{16} - 16 a^{15} + 66 a^{14} - 182 a^{13} + 371 a^{12} - 588 a^{11} + 745 a^{10} - 766 a^{9} + 647 a^{8} - 456 a^{7} + 279 a^{6} - 158 a^{5} + 83 a^{4} - 36 a^{3} + 13 a^{2} - 4 a \),  \( a^{14} - 7 a^{13} + 26 a^{12} - 65 a^{11} + 121 a^{10} - 176 a^{9} + 206 a^{8} - 197 a^{7} + 157 a^{6} - 106 a^{5} + 64 a^{4} - 35 a^{3} + 17 a^{2} - 6 a + 3 \),  \( a \),  \( a - 1 \),  \( \frac{2}{7} a^{17} - \frac{24}{7} a^{16} + \frac{128}{7} a^{15} - \frac{431}{7} a^{14} + \frac{1038}{7} a^{13} - \frac{1903}{7} a^{12} + \frac{2742}{7} a^{11} - \frac{3152}{7} a^{10} + \frac{2893}{7} a^{9} - \frac{2102}{7} a^{8} + \frac{1188}{7} a^{7} - \frac{526}{7} a^{6} + \frac{204}{7} a^{5} - \frac{85}{7} a^{4} + \frac{33}{7} a^{3} - \frac{19}{7} a^{2} + \frac{1}{7} a + \frac{3}{7} \),  \( \frac{19}{7} a^{17} - \frac{165}{7} a^{16} + \frac{740}{7} a^{15} - \frac{2229}{7} a^{14} + \frac{4996}{7} a^{13} - \frac{8779}{7} a^{12} + \frac{12462}{7} a^{11} - \frac{14544}{7} a^{10} + \frac{14124}{7} a^{9} - \frac{11527}{7} a^{8} + \frac{8031}{7} a^{7} - \frac{4906}{7} a^{6} + \frac{2701}{7} a^{5} - \frac{1336}{7} a^{4} + \frac{583}{7} a^{3} - \frac{212}{7} a^{2} + \frac{62}{7} a - \frac{10}{7} \),  \( \frac{20}{7} a^{17} - \frac{163}{7} a^{16} + \frac{692}{7} a^{15} - \frac{1986}{7} a^{14} + \frac{4269}{7} a^{13} - \frac{7249}{7} a^{12} + \frac{10032}{7} a^{11} - \frac{11535}{7} a^{10} + \frac{11178}{7} a^{9} - \frac{9232}{7} a^{8} + \frac{6595}{7} a^{7} - \frac{4147}{7} a^{6} + \frac{2313}{7} a^{5} - \frac{1144}{7} a^{4} + \frac{526}{7} a^{3} - \frac{218}{7} a^{2} + \frac{66}{7} a - \frac{19}{7} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 61.0464139366 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T968:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 185794560
The 300 conjugacy class representatives for t18n968 are not computed
Character table for t18n968 is not computed

Intermediate fields

9.1.53628353.1

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$
17.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
43.6.0.1$x^{6} - x + 26$$1$$6$$0$$C_6$$[\ ]^{6}$
2311Data not computed
73363Data not computed