# Properties

 Label 15.3.19664802801826816.1 Degree $15$ Signature $[3, 6]$ Discriminant $2^{10}\cdot 79^{7}$ Root discriminant $12.20$ Ramified primes $2, 79$ Class number $1$ Class group Trivial Galois Group $D_5\times S_3$ (as 15T7)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 21, -16, -18, 50, -39, -10, 47, -41, 7, 19, -23, 14, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 14*x^13 - 23*x^12 + 19*x^11 + 7*x^10 - 41*x^9 + 47*x^8 - 10*x^7 - 39*x^6 + 50*x^5 - 18*x^4 - 16*x^3 + 21*x^2 - 9*x + 1)
gp: K = bnfinit(x^15 - 5*x^14 + 14*x^13 - 23*x^12 + 19*x^11 + 7*x^10 - 41*x^9 + 47*x^8 - 10*x^7 - 39*x^6 + 50*x^5 - 18*x^4 - 16*x^3 + 21*x^2 - 9*x + 1, 1)

## Normalizeddefining polynomial

$$x^{15}$$ $$\mathstrut -\mathstrut 5 x^{14}$$ $$\mathstrut +\mathstrut 14 x^{13}$$ $$\mathstrut -\mathstrut 23 x^{12}$$ $$\mathstrut +\mathstrut 19 x^{11}$$ $$\mathstrut +\mathstrut 7 x^{10}$$ $$\mathstrut -\mathstrut 41 x^{9}$$ $$\mathstrut +\mathstrut 47 x^{8}$$ $$\mathstrut -\mathstrut 10 x^{7}$$ $$\mathstrut -\mathstrut 39 x^{6}$$ $$\mathstrut +\mathstrut 50 x^{5}$$ $$\mathstrut -\mathstrut 18 x^{4}$$ $$\mathstrut -\mathstrut 16 x^{3}$$ $$\mathstrut +\mathstrut 21 x^{2}$$ $$\mathstrut -\mathstrut 9 x$$ $$\mathstrut +\mathstrut 1$$

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

## Invariants

 Degree: $15$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[3, 6]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$19664802801826816=2^{10}\cdot 79^{7}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $12.20$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 79$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} - \frac{8}{17} a^{13} + \frac{4}{17} a^{12} - \frac{1}{17} a^{11} + \frac{5}{17} a^{10} - \frac{8}{17} a^{9} - \frac{4}{17} a^{7} + \frac{2}{17} a^{6} + \frac{6}{17} a^{5} - \frac{2}{17} a^{4} + \frac{5}{17} a^{3} + \frac{3}{17} a^{2} - \frac{5}{17} a + \frac{6}{17}$

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: $$\frac{13}{17} a^{14} - \frac{53}{17} a^{13} + \frac{120}{17} a^{12} - \frac{132}{17} a^{11} - \frac{3}{17} a^{10} + \frac{236}{17} a^{9} - 19 a^{8} + \frac{67}{17} a^{7} + \frac{298}{17} a^{6} - \frac{364}{17} a^{5} + \frac{59}{17} a^{4} + \frac{201}{17} a^{3} - \frac{148}{17} a^{2} - \frac{14}{17} a + \frac{27}{17}$$,  $$\frac{14}{17} a^{14} - \frac{61}{17} a^{13} + \frac{158}{17} a^{12} - \frac{235}{17} a^{11} + \frac{155}{17} a^{10} + \frac{126}{17} a^{9} - 26 a^{8} + \frac{403}{17} a^{7} + \frac{11}{17} a^{6} - \frac{443}{17} a^{5} + \frac{448}{17} a^{4} - \frac{66}{17} a^{3} - \frac{196}{17} a^{2} + \frac{185}{17} a - \frac{35}{17}$$,  $$\frac{1}{17} a^{14} - \frac{8}{17} a^{13} + \frac{21}{17} a^{12} - \frac{35}{17} a^{11} + \frac{22}{17} a^{10} + \frac{26}{17} a^{9} - 4 a^{8} + \frac{64}{17} a^{7} + \frac{19}{17} a^{6} - \frac{62}{17} a^{5} + \frac{49}{17} a^{4} + \frac{5}{17} a^{3} - \frac{31}{17} a^{2} + \frac{12}{17} a - \frac{11}{17}$$,  $$\frac{5}{17} a^{14} - \frac{6}{17} a^{13} - \frac{14}{17} a^{12} + \frac{80}{17} a^{11} - \frac{145}{17} a^{10} + \frac{79}{17} a^{9} + 8 a^{8} - \frac{326}{17} a^{7} + \frac{197}{17} a^{6} + \frac{166}{17} a^{5} - \frac{350}{17} a^{4} + \frac{161}{17} a^{3} + \frac{117}{17} a^{2} - \frac{144}{17} a + \frac{30}{17}$$,  $$\frac{23}{17} a^{14} - \frac{99}{17} a^{13} + \frac{245}{17} a^{12} - \frac{329}{17} a^{11} + \frac{149}{17} a^{10} + \frac{309}{17} a^{9} - 40 a^{8} + \frac{452}{17} a^{7} + \frac{216}{17} a^{6} - \frac{678}{17} a^{5} + \frac{447}{17} a^{4} + \frac{64}{17} a^{3} - \frac{271}{17} a^{2} + \frac{157}{17} a - \frac{32}{17}$$,  $$\frac{7}{17} a^{14} - \frac{22}{17} a^{13} + \frac{45}{17} a^{12} - \frac{41}{17} a^{11} - \frac{16}{17} a^{10} + \frac{97}{17} a^{9} - 8 a^{8} + \frac{40}{17} a^{7} + \frac{99}{17} a^{6} - \frac{162}{17} a^{5} + \frac{71}{17} a^{4} + \frac{69}{17} a^{3} - \frac{98}{17} a^{2} + \frac{33}{17} a + \frac{8}{17}$$,  $$\frac{14}{17} a^{14} - \frac{61}{17} a^{13} + \frac{158}{17} a^{12} - \frac{235}{17} a^{11} + \frac{155}{17} a^{10} + \frac{126}{17} a^{9} - 26 a^{8} + \frac{403}{17} a^{7} + \frac{11}{17} a^{6} - \frac{443}{17} a^{5} + \frac{448}{17} a^{4} - \frac{66}{17} a^{3} - \frac{196}{17} a^{2} + \frac{185}{17} a - \frac{52}{17}$$,  $$\frac{13}{17} a^{14} - \frac{53}{17} a^{13} + \frac{120}{17} a^{12} - \frac{132}{17} a^{11} - \frac{3}{17} a^{10} + \frac{236}{17} a^{9} - 19 a^{8} + \frac{67}{17} a^{7} + \frac{298}{17} a^{6} - \frac{364}{17} a^{5} + \frac{59}{17} a^{4} + \frac{201}{17} a^{3} - \frac{148}{17} a^{2} + \frac{3}{17} a + \frac{27}{17}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$103.698212623$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$S_3\times D_5$ (as 15T7):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 60 The 12 conjugacy class representatives for $D_5\times S_3$ Character table for $D_5\times S_3$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling fields

 Degree 30 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5} 2.10.10.7x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
79Data not computed