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Properties

Label 18.2.30568999800...5653.1
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 13^{5}\cdot 29^{6}$
Root discriminant $22.92$
Ramified primes $7, 13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T705

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 16, -9, -298, 133, 838, -1229, 345, 722, -958, 498, 20, -227, 165, -37, -22, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 21*x^16 - 22*x^15 - 37*x^14 + 165*x^13 - 227*x^12 + 20*x^11 + 498*x^10 - 958*x^9 + 722*x^8 + 345*x^7 - 1229*x^6 + 838*x^5 + 133*x^4 - 298*x^3 - 9*x^2 + 16*x + 29)
 
gp: K = bnfinit(x^18 - 7*x^17 + 21*x^16 - 22*x^15 - 37*x^14 + 165*x^13 - 227*x^12 + 20*x^11 + 498*x^10 - 958*x^9 + 722*x^8 + 345*x^7 - 1229*x^6 + 838*x^5 + 133*x^4 - 298*x^3 - 9*x^2 + 16*x + 29, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 21 x^{16} - 22 x^{15} - 37 x^{14} + 165 x^{13} - 227 x^{12} + 20 x^{11} + 498 x^{10} - 958 x^{9} + 722 x^{8} + 345 x^{7} - 1229 x^{6} + 838 x^{5} + 133 x^{4} - 298 x^{3} - 9 x^{2} + 16 x + 29 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3056899980033856376345653=7^{12}\cdot 13^{5}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.92$
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:  $7, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13} a^{15} + \frac{5}{13} a^{14} + \frac{5}{13} a^{13} + \frac{2}{13} a^{12} + \frac{1}{13} a^{11} + \frac{3}{13} a^{10} + \frac{5}{13} a^{9} + \frac{1}{13} a^{8} + \frac{5}{13} a^{7} + \frac{5}{13} a^{6} + \frac{5}{13} a^{5} + \frac{3}{13} a^{4} + \frac{4}{13} a^{3} - \frac{1}{13} a^{2} + \frac{4}{13} a + \frac{6}{13}$, $\frac{1}{13} a^{16} + \frac{6}{13} a^{14} + \frac{3}{13} a^{13} + \frac{4}{13} a^{12} - \frac{2}{13} a^{11} + \frac{3}{13} a^{10} + \frac{2}{13} a^{9} + \frac{6}{13} a^{7} + \frac{6}{13} a^{6} + \frac{4}{13} a^{5} + \frac{2}{13} a^{4} + \frac{5}{13} a^{3} - \frac{4}{13} a^{2} - \frac{1}{13} a - \frac{4}{13}$, $\frac{1}{187275233799397757} a^{17} + \frac{3882429311851469}{187275233799397757} a^{16} - \frac{2175841988538106}{187275233799397757} a^{15} - \frac{19769837356792381}{187275233799397757} a^{14} + \frac{28636617634431996}{187275233799397757} a^{13} - \frac{5780452889402974}{187275233799397757} a^{12} - \frac{62774115263181386}{187275233799397757} a^{11} - \frac{4090519614264745}{187275233799397757} a^{10} + \frac{1303735697557502}{187275233799397757} a^{9} + \frac{44721548460653904}{187275233799397757} a^{8} + \frac{85740776982383213}{187275233799397757} a^{7} + \frac{6848095857701292}{14405787215338289} a^{6} - \frac{61546928016137622}{187275233799397757} a^{5} + \frac{40877372231778799}{187275233799397757} a^{4} + \frac{61057815896433864}{187275233799397757} a^{3} - \frac{41371077079458150}{187275233799397757} a^{2} + \frac{90420318006798587}{187275233799397757} a + \frac{79881694153930543}{187275233799397757}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 82213.8752039 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T705:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 64 conjugacy class representatives for t18n705 are not computed
Character table for t18n705 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.5.16721334721.1

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ R $18$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$