# Properties

 Label 18.2.48373192518...4741.1 Degree $18$ Signature $[2, 8]$ Discriminant $23^{6}\cdot 59^{2}\cdot 149\cdot 251^{2}$ Root discriminant $10.92$ Ramified primes $23, 59, 149, 251$ Class number $1$ Class group Trivial Galois Group 18T912

# Learn more about

Show commands for: Magma / SageMath / Pari/GP

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

## Normalizeddefining polynomial

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

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

## Invariants

 Degree: $18$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[2, 8]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$4837319251866194741=23^{6}\cdot 59^{2}\cdot 149\cdot 251^{2}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $10.92$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $23, 59, 149, 251$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{649} a^{16} - \frac{153}{649} a^{15} + \frac{4}{11} a^{14} - \frac{206}{649} a^{13} + \frac{16}{59} a^{12} - \frac{240}{649} a^{11} + \frac{131}{649} a^{10} + \frac{31}{649} a^{9} - \frac{221}{649} a^{8} + \frac{31}{649} a^{7} + \frac{131}{649} a^{6} - \frac{240}{649} a^{5} + \frac{16}{59} a^{4} - \frac{206}{649} a^{3} + \frac{4}{11} a^{2} - \frac{153}{649} a + \frac{1}{649}$, $\frac{1}{649} a^{17} + \frac{191}{649} a^{15} + \frac{207}{649} a^{14} - \frac{190}{649} a^{13} + \frac{79}{649} a^{12} - \frac{245}{649} a^{11} - \frac{45}{649} a^{10} - \frac{21}{649} a^{9} - \frac{34}{649} a^{8} - \frac{318}{649} a^{7} - \frac{316}{649} a^{6} - \frac{200}{649} a^{5} + \frac{113}{649} a^{4} - \frac{130}{649} a^{3} + \frac{260}{649} a^{2} - \frac{4}{59} a + \frac{153}{649}$

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: $9$ 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: $$\frac{1143}{59} a^{17} - \frac{2806}{59} a^{16} + \frac{755}{59} a^{15} - \frac{5299}{59} a^{14} + \frac{14241}{59} a^{13} + \frac{945}{59} a^{12} + \frac{2766}{59} a^{11} - \frac{31568}{59} a^{10} + \frac{2055}{59} a^{9} + \frac{13743}{59} a^{8} + \frac{26850}{59} a^{7} - \frac{18178}{59} a^{6} - \frac{7690}{59} a^{5} - \frac{11110}{59} a^{4} + \frac{10841}{59} a^{3} + \frac{351}{59} a^{2} + \frac{2488}{59} a - \frac{1977}{59}$$,  $$a$$,  $$\frac{7134}{649} a^{17} - \frac{17297}{649} a^{16} + \frac{4056}{649} a^{15} - \frac{32718}{649} a^{14} + \frac{8008}{59} a^{13} + \frac{8878}{649} a^{12} + \frac{16428}{649} a^{11} - \frac{197968}{649} a^{10} + \frac{7760}{649} a^{9} + \frac{89110}{649} a^{8} + \frac{172144}{649} a^{7} - \frac{112892}{649} a^{6} - \frac{4958}{59} a^{5} - \frac{69172}{649} a^{4} + \frac{69616}{649} a^{3} + \frac{4010}{649} a^{2} + \frac{14966}{649} a - \frac{1170}{59}$$,  $$\frac{13470}{649} a^{17} - \frac{32492}{649} a^{16} + \frac{7858}{649} a^{15} - \frac{62944}{649} a^{14} + \frac{165417}{649} a^{13} + \frac{16391}{649} a^{12} + \frac{37353}{649} a^{11} - \frac{370224}{649} a^{10} + \frac{10474}{649} a^{9} + \frac{154223}{649} a^{8} + \frac{323138}{649} a^{7} - \frac{199931}{649} a^{6} - \frac{93112}{649} a^{5} - \frac{138285}{649} a^{4} + \frac{123427}{649} a^{3} + \frac{5860}{649} a^{2} + \frac{30945}{649} a - \frac{23071}{649}$$,  $$\frac{6705}{649} a^{17} - \frac{16760}{649} a^{16} + \frac{4802}{649} a^{15} - \frac{30484}{649} a^{14} + \frac{84936}{649} a^{13} + \frac{3301}{649} a^{12} + \frac{12123}{649} a^{11} - \frac{187494}{649} a^{10} + \frac{19787}{649} a^{9} + \frac{88859}{649} a^{8} + \frac{155824}{649} a^{7} - \frac{117906}{649} a^{6} - \frac{47645}{649} a^{5} - \frac{60130}{649} a^{4} + \frac{67982}{649} a^{3} + \frac{1679}{649} a^{2} + \frac{13985}{649} a - \frac{12421}{649}$$,  $$\frac{8175}{649} a^{17} - \frac{19983}{649} a^{16} + \frac{5083}{649} a^{15} - \frac{37714}{649} a^{14} + \frac{101591}{649} a^{13} + \frac{8430}{649} a^{12} + \frac{19219}{649} a^{11} - \frac{226749}{649} a^{10} + \frac{11016}{649} a^{9} + \frac{100215}{649} a^{8} + \frac{195915}{649} a^{7} - \frac{127191}{649} a^{6} - \frac{58769}{649} a^{5} - \frac{82252}{649} a^{4} + \frac{76785}{649} a^{3} + \frac{4863}{649} a^{2} + \frac{18627}{649} a - \frac{13990}{649}$$,  $$\frac{4869}{649} a^{17} - \frac{12306}{649} a^{16} + \frac{3925}{649} a^{15} - \frac{22670}{649} a^{14} + \frac{62710}{649} a^{13} + \frac{300}{649} a^{12} + \frac{10831}{649} a^{11} - \frac{137950}{649} a^{10} + \frac{17293}{649} a^{9} + \frac{60622}{649} a^{8} + \frac{115171}{649} a^{7} - \frac{86112}{649} a^{6} - \frac{31611}{649} a^{5} - \frac{47026}{649} a^{4} + \frac{51767}{649} a^{3} + \frac{449}{649} a^{2} + \frac{11685}{649} a - \frac{9805}{649}$$,  $$\frac{2865}{649} a^{17} - \frac{6697}{649} a^{16} + \frac{116}{59} a^{15} - \frac{1208}{59} a^{14} + \frac{33717}{649} a^{13} + \frac{5587}{649} a^{12} + 13 a^{11} - \frac{75566}{649} a^{10} - \frac{2331}{649} a^{9} + \frac{30760}{649} a^{8} + \frac{66397}{649} a^{7} - \frac{37486}{649} a^{6} - \frac{18398}{649} a^{5} - \frac{29399}{649} a^{4} + \frac{21950}{649} a^{3} + \frac{1618}{649} a^{2} + \frac{6206}{649} a - \frac{4480}{649}$$,  $$\frac{8663}{649} a^{17} - \frac{1916}{59} a^{16} + \frac{5271}{649} a^{15} - \frac{40184}{649} a^{14} + \frac{107474}{649} a^{13} + \frac{9076}{649} a^{12} + \frac{21795}{649} a^{11} - \frac{240675}{649} a^{10} + \frac{11666}{649} a^{9} + \frac{104516}{649} a^{8} + \frac{208687}{649} a^{7} - \frac{136426}{649} a^{6} - \frac{62789}{649} a^{5} - \frac{85782}{649} a^{4} + \frac{84041}{649} a^{3} + \frac{3595}{649} a^{2} + \frac{1728}{59} a - \frac{15703}{649}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$67.155663406$$ 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 663552 The 330 conjugacy class representatives for t18n912 are not computed Character table for t18n912 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 $18$ $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ R

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$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2} 23.2.0.1x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2} 23.4.2.1x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4} 59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 59.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 59.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 59.4.2.1x^{4} + 177 x^{2} + 13924$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
149Data not computed
251Data not computed