Properties

Label 18.0.11860120711...0699.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{4}\cdot 139\cdot 1373^{4}$
Root discriminant $10.10$
Ramified primes $7, 139, 1373$
Class number $1$
Class group Trivial
Galois Group 18T966

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 38, -101, 195, -300, 395, -463, 489, -465, 400, -312, 221, -141, 80, -39, 16, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 16*x^16 - 39*x^15 + 80*x^14 - 141*x^13 + 221*x^12 - 312*x^11 + 400*x^10 - 465*x^9 + 489*x^8 - 463*x^7 + 395*x^6 - 300*x^5 + 195*x^4 - 101*x^3 + 38*x^2 - 9*x + 1)
gp: K = bnfinit(x^18 - 5*x^17 + 16*x^16 - 39*x^15 + 80*x^14 - 141*x^13 + 221*x^12 - 312*x^11 + 400*x^10 - 465*x^9 + 489*x^8 - 463*x^7 + 395*x^6 - 300*x^5 + 195*x^4 - 101*x^3 + 38*x^2 - 9*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 5 x^{17} \) \(\mathstrut +\mathstrut 16 x^{16} \) \(\mathstrut -\mathstrut 39 x^{15} \) \(\mathstrut +\mathstrut 80 x^{14} \) \(\mathstrut -\mathstrut 141 x^{13} \) \(\mathstrut +\mathstrut 221 x^{12} \) \(\mathstrut -\mathstrut 312 x^{11} \) \(\mathstrut +\mathstrut 400 x^{10} \) \(\mathstrut -\mathstrut 465 x^{9} \) \(\mathstrut +\mathstrut 489 x^{8} \) \(\mathstrut -\mathstrut 463 x^{7} \) \(\mathstrut +\mathstrut 395 x^{6} \) \(\mathstrut -\mathstrut 300 x^{5} \) \(\mathstrut +\mathstrut 195 x^{4} \) \(\mathstrut -\mathstrut 101 x^{3} \) \(\mathstrut +\mathstrut 38 x^{2} \) \(\mathstrut -\mathstrut 9 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-1186012071116620699=-\,7^{4}\cdot 139\cdot 1373^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.10$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 139, 1373$
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 \(a\))

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

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:  \( 22.1660529321 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T966:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 92897280
The 168 conjugacy class representatives for t18n966 are not computed
Character table for t18n966 is not computed

Intermediate fields

9.1.92371321.1

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 $18$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.5.0.1$x^{5} - x + 4$$1$$5$$0$$C_5$$[\ ]^{5}$
7.5.0.1$x^{5} - x + 4$$1$$5$$0$$C_5$$[\ ]^{5}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$
139.7.0.1$x^{7} - 2 x + 4$$1$$7$$0$$C_7$$[\ ]^{7}$
139.7.0.1$x^{7} - 2 x + 4$$1$$7$$0$$C_7$$[\ ]^{7}$
1373Data not computed