↑

Properties

Label 15.11.2264485717...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{12}\cdot 5^{8}\cdot 13^{10}\cdot 101323^{2}$
Root discriminant $105.60$
Ramified primes $2, 5, 13, 101323$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T50

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![235, -4915, 1177, 36939, 55003, 2237, -83815, 1339, 31833, -3601, -4717, 825, 289, -57, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 57*x^13 + 289*x^12 + 825*x^11 - 4717*x^10 - 3601*x^9 + 31833*x^8 + 1339*x^7 - 83815*x^6 + 2237*x^5 + 55003*x^4 + 36939*x^3 + 1177*x^2 - 4915*x + 235)
 
gp: K = bnfinit(x^15 - 5*x^14 - 57*x^13 + 289*x^12 + 825*x^11 - 4717*x^10 - 3601*x^9 + 31833*x^8 + 1339*x^7 - 83815*x^6 + 2237*x^5 + 55003*x^4 + 36939*x^3 + 1177*x^2 - 4915*x + 235, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 57 x^{13} + 289 x^{12} + 825 x^{11} - 4717 x^{10} - 3601 x^{9} + 31833 x^{8} + 1339 x^{7} - 83815 x^{6} + 2237 x^{5} + 55003 x^{4} + 36939 x^{3} + 1177 x^{2} - 4915 x + 235 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2264485717039079949313600000000=2^{12}\cdot 5^{8}\cdot 13^{10}\cdot 101323^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.60$
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:  $2, 5, 13, 101323$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{1}{16} a^{3} + \frac{1}{16} a^{2} - \frac{7}{16} a + \frac{7}{16}$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{16} a^{2} - \frac{3}{8} a - \frac{5}{32}$, $\frac{1}{128} a^{9} + \frac{1}{128} a^{8} + \frac{1}{64} a^{7} + \frac{3}{64} a^{6} - \frac{3}{32} a^{5} - \frac{1}{8} a^{4} + \frac{11}{64} a^{3} + \frac{1}{64} a^{2} + \frac{51}{128} a - \frac{57}{128}$, $\frac{1}{256} a^{10} + \frac{1}{256} a^{8} + \frac{1}{64} a^{7} + \frac{7}{128} a^{6} - \frac{1}{64} a^{5} + \frac{3}{128} a^{4} + \frac{11}{64} a^{3} - \frac{47}{256} a^{2} - \frac{11}{64} a + \frac{25}{256}$, $\frac{1}{1024} a^{11} - \frac{1}{1024} a^{10} + \frac{1}{1024} a^{9} - \frac{13}{1024} a^{8} + \frac{5}{512} a^{7} - \frac{25}{512} a^{6} - \frac{27}{512} a^{5} + \frac{19}{512} a^{4} + \frac{165}{1024} a^{3} + \frac{163}{1024} a^{2} - \frac{379}{1024} a + \frac{119}{1024}$, $\frac{1}{2048} a^{12} + \frac{1}{512} a^{9} + \frac{13}{2048} a^{8} - \frac{1}{256} a^{7} - \frac{1}{256} a^{6} - \frac{13}{128} a^{5} - \frac{53}{2048} a^{4} + \frac{21}{256} a^{3} + \frac{41}{256} a^{2} + \frac{11}{512} a - \frac{281}{2048}$, $\frac{1}{4096} a^{13} - \frac{1}{4096} a^{12} + \frac{1}{1024} a^{10} + \frac{9}{4096} a^{9} - \frac{21}{4096} a^{8} - \frac{25}{512} a^{6} - \frac{357}{4096} a^{5} - \frac{291}{4096} a^{4} + \frac{5}{128} a^{3} - \frac{71}{1024} a^{2} - \frac{1861}{4096} a - \frac{1255}{4096}$, $\frac{1}{5780610580480} a^{14} + \frac{18864159}{2890305290240} a^{13} - \frac{1204108623}{5780610580480} a^{12} - \frac{93484539}{289030529024} a^{11} - \frac{703117087}{1156122116096} a^{10} + \frac{8468526049}{2890305290240} a^{9} + \frac{88035529573}{5780610580480} a^{8} - \frac{21800725201}{722576322560} a^{7} - \frac{65535793529}{1156122116096} a^{6} + \frac{14716645549}{578061058048} a^{5} - \frac{63608991213}{5780610580480} a^{4} - \frac{216044613399}{1445152645120} a^{3} + \frac{1090466500151}{5780610580480} a^{2} - \frac{241092546669}{578061058048} a - \frac{375358461717}{1156122116096}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $12$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28726471148.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T50:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3000
The 32 conjugacy class representatives for [D(5)^3]3=D(5)wr3
Character table for [D(5)^3]3=D(5)wr3 is not computed

Intermediate fields

3.3.169.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ $15$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.5.6.1$x^{5} + 10 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
13Data not computed
101323Data not computed