# Properties

 Label 18.0.14172006159...9707.1 Degree $18$ Signature $[0, 9]$ Discriminant $-\,23^{6}\cdot 43^{3}\cdot 347^{2}$ Root discriminant $10.20$ Ramified primes $23, 43, 347$ Class number $1$ Class group Trivial Galois Group 18T556

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 13, -23, 30, -32, 31, -32, 36, -38, 36, -29, 21, -15, 12, -9, 6, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 9*x^15 + 12*x^14 - 15*x^13 + 21*x^12 - 29*x^11 + 36*x^10 - 38*x^9 + 36*x^8 - 32*x^7 + 31*x^6 - 32*x^5 + 30*x^4 - 23*x^3 + 13*x^2 - 5*x + 1)
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 - 9*x^15 + 12*x^14 - 15*x^13 + 21*x^12 - 29*x^11 + 36*x^10 - 38*x^9 + 36*x^8 - 32*x^7 + 31*x^6 - 32*x^5 + 30*x^4 - 23*x^3 + 13*x^2 - 5*x + 1, 1)

## Normalizeddefining polynomial

$$x^{18}$$ $$\mathstrut -\mathstrut 3 x^{17}$$ $$\mathstrut +\mathstrut 6 x^{16}$$ $$\mathstrut -\mathstrut 9 x^{15}$$ $$\mathstrut +\mathstrut 12 x^{14}$$ $$\mathstrut -\mathstrut 15 x^{13}$$ $$\mathstrut +\mathstrut 21 x^{12}$$ $$\mathstrut -\mathstrut 29 x^{11}$$ $$\mathstrut +\mathstrut 36 x^{10}$$ $$\mathstrut -\mathstrut 38 x^{9}$$ $$\mathstrut +\mathstrut 36 x^{8}$$ $$\mathstrut -\mathstrut 32 x^{7}$$ $$\mathstrut +\mathstrut 31 x^{6}$$ $$\mathstrut -\mathstrut 32 x^{5}$$ $$\mathstrut +\mathstrut 30 x^{4}$$ $$\mathstrut -\mathstrut 23 x^{3}$$ $$\mathstrut +\mathstrut 13 x^{2}$$ $$\mathstrut -\mathstrut 5 x$$ $$\mathstrut +\mathstrut 1$$

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: $$-1417200615982289707=-\,23^{6}\cdot 43^{3}\cdot 347^{2}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $10.20$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $23, 43, 347$ 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 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: $$26 a^{17} - 60 a^{16} + 115 a^{15} - 155 a^{14} + 206 a^{13} - 249 a^{12} + 376 a^{11} - 496 a^{10} + 597 a^{9} - 580 a^{8} + 541 a^{7} - 463 a^{6} + 491 a^{5} - 496 a^{4} + 442 a^{3} - 298 a^{2} + 137 a - 38$$,  $$a$$,  $$38 a^{17} - 89 a^{16} + 170 a^{15} - 230 a^{14} + 304 a^{13} - 368 a^{12} + 554 a^{11} - 735 a^{10} + 883 a^{9} - 858 a^{8} + 796 a^{7} - 683 a^{6} + 722 a^{5} - 735 a^{4} + 653 a^{3} - 438 a^{2} + 198 a - 54$$,  $$28 a^{17} - 63 a^{16} + 119 a^{15} - 160 a^{14} + 212 a^{13} - 256 a^{12} + 389 a^{11} - 512 a^{10} + 610 a^{9} - 590 a^{8} + 549 a^{7} - 470 a^{6} + 501 a^{5} - 508 a^{4} + 445 a^{3} - 297 a^{2} + 133 a - 36$$,  $$47 a^{17} - 109 a^{16} + 207 a^{15} - 279 a^{14} + 369 a^{13} - 447 a^{12} + 675 a^{11} - 893 a^{10} + 1069 a^{9} - 1036 a^{8} + 962 a^{7} - 826 a^{6} + 876 a^{5} - 889 a^{4} + 785 a^{3} - 525 a^{2} + 237 a - 64$$,  $$13 a^{17} - 29 a^{16} + 56 a^{15} - 75 a^{14} + 100 a^{13} - 120 a^{12} + 183 a^{11} - 240 a^{10} + 289 a^{9} - 279 a^{8} + 261 a^{7} - 222 a^{6} + 238 a^{5} - 240 a^{4} + 213 a^{3} - 141 a^{2} + 64 a - 18$$,  $$26 a^{17} - 58 a^{16} + 112 a^{15} - 150 a^{14} + 200 a^{13} - 241 a^{12} + 366 a^{11} - 479 a^{10} + 577 a^{9} - 559 a^{8} + 522 a^{7} - 445 a^{6} + 475 a^{5} - 477 a^{4} + 424 a^{3} - 286 a^{2} + 129 a - 36$$,  $$11 a^{17} - 27 a^{16} + 52 a^{15} - 71 a^{14} + 94 a^{13} - 114 a^{12} + 170 a^{11} - 227 a^{10} + 275 a^{9} - 269 a^{8} + 250 a^{7} - 215 a^{6} + 225 a^{5} - 229 a^{4} + 207 a^{3} - 140 a^{2} + 66 a - 18$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$24.7750980478$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 10368 The 98 conjugacy class representatives for t18n556 are not computed Character table for t18n556 is not computed

## Intermediate fields

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $18$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$23$$\Q_{23}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{23}$$x + 2$$1$$1$$0Trivial[\ ] 23.2.0.1x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2} 23.6.3.2x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3} 4343.2.1.2x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2} 43.2.0.1x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4} 43.4.2.1x^{4} + 215 x^{2} + 16641$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
347Data not computed