# 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

# Related objects

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

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

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(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $10.92$ 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: $23, 59, 149, 251$ 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 $$a$$)

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

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

Trivial 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: $9$ 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 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$$C_2^2$$[\ ]_{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$$C_2^2$$[\ ]_{2}^{2}$
149Data not computed
251Data not computed