↑

Properties

Label 15.15.1043423272...1584.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{6}\cdot 3^{20}\cdot 881^{6}$
Root discriminant $86.01$
Ramified primes $2, 3, 881$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T53

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4096, -39936, 191232, 405952, -114816, -349488, 28976, 112068, -3456, -17513, 192, 1440, -4, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 60*x^13 - 4*x^12 + 1440*x^11 + 192*x^10 - 17513*x^9 - 3456*x^8 + 112068*x^7 + 28976*x^6 - 349488*x^5 - 114816*x^4 + 405952*x^3 + 191232*x^2 - 39936*x - 4096)
 
gp: K = bnfinit(x^15 - 60*x^13 - 4*x^12 + 1440*x^11 + 192*x^10 - 17513*x^9 - 3456*x^8 + 112068*x^7 + 28976*x^6 - 349488*x^5 - 114816*x^4 + 405952*x^3 + 191232*x^2 - 39936*x - 4096, 1)
 

Normalized defining polynomial

\( x^{15} - 60 x^{13} - 4 x^{12} + 1440 x^{11} + 192 x^{10} - 17513 x^{9} - 3456 x^{8} + 112068 x^{7} + 28976 x^{6} - 349488 x^{5} - 114816 x^{4} + 405952 x^{3} + 191232 x^{2} - 39936 x - 4096 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(104342327215448868113348971584=2^{6}\cdot 3^{20}\cdot 881^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.01$
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:  $2, 3, 881$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{6} a$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{2}$, $\frac{1}{48} a^{9} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{23}{48} a^{3} - \frac{5}{12} a + \frac{1}{3}$, $\frac{1}{96} a^{10} - \frac{1}{24} a^{8} - \frac{1}{24} a^{7} + \frac{23}{96} a^{4} - \frac{1}{2} a^{3} - \frac{5}{24} a^{2} + \frac{1}{6} a$, $\frac{1}{384} a^{11} - \frac{1}{192} a^{10} - \frac{1}{96} a^{9} + \frac{1}{96} a^{8} - \frac{1}{48} a^{7} + \frac{23}{384} a^{5} - \frac{23}{192} a^{4} + \frac{19}{96} a^{3} + \frac{19}{48} a^{2} + \frac{1}{12} a$, $\frac{1}{62208} a^{12} - \frac{1}{1296} a^{10} - \frac{5}{1728} a^{9} + \frac{1}{72} a^{8} - \frac{1}{16} a^{7} + \frac{3799}{62208} a^{6} - \frac{1}{4} a^{5} + \frac{521}{2592} a^{4} + \frac{11}{216} a^{3} - \frac{89}{432} a^{2} - \frac{4}{9} a + \frac{103}{243}$, $\frac{1}{248832} a^{13} - \frac{1}{5184} a^{11} - \frac{5}{6912} a^{10} + \frac{1}{288} a^{9} - \frac{1}{64} a^{8} + \frac{3799}{248832} a^{7} + \frac{1}{48} a^{6} + \frac{521}{10368} a^{5} + \frac{11}{864} a^{4} + \frac{343}{1728} a^{3} + \frac{7}{18} a^{2} - \frac{35}{243} a - \frac{1}{3}$, $\frac{1}{995328} a^{14} - \frac{1}{497664} a^{13} - \frac{1}{124416} a^{12} - \frac{7}{82944} a^{11} + \frac{187}{41472} a^{10} - \frac{17}{6912} a^{9} - \frac{16073}{995328} a^{8} + \frac{14345}{497664} a^{7} - \frac{737}{62208} a^{6} + \frac{2137}{20736} a^{5} + \frac{3433}{20736} a^{4} - \frac{899}{3456} a^{3} + \frac{3437}{7776} a^{2} - \frac{481}{1944} a + \frac{191}{486}$

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:  $14$
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:  \( 104325761724 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T53:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 4860
The 24 conjugacy class representatives for [3^4]A(5)
Character table for [3^4]A(5) is not computed

Intermediate fields

5.5.3104644.1

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

Sibling fields

Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 45 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 R ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.6.8.9$x^{6} + 6 x^{5} + 9$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
881Data not computed