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)

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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)

Normalized defining 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

3.3.316.1, 5.1.6241.1

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type 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$$[\ ]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
79Data not computed