↑

Properties

Label 15.3.55507379263...3504.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{12}\cdot 7^{10}\cdot 13^{2}\cdot 883^{2}\cdot 19081^{2}$
Root discriminant $82.47$
Ramified primes $2, 7, 13, 883, 19081$
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![2107, -13475, -9583, 7603, 69219, 11733, -40119, -6709, 13241, 815, -2325, 65, 233, -33, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 33*x^13 + 233*x^12 + 65*x^11 - 2325*x^10 + 815*x^9 + 13241*x^8 - 6709*x^7 - 40119*x^6 + 11733*x^5 + 69219*x^4 + 7603*x^3 - 9583*x^2 - 13475*x + 2107)
 
gp: K = bnfinit(x^15 - 5*x^14 - 33*x^13 + 233*x^12 + 65*x^11 - 2325*x^10 + 815*x^9 + 13241*x^8 - 6709*x^7 - 40119*x^6 + 11733*x^5 + 69219*x^4 + 7603*x^3 - 9583*x^2 - 13475*x + 2107, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 33 x^{13} + 233 x^{12} + 65 x^{11} - 2325 x^{10} + 815 x^{9} + 13241 x^{8} - 6709 x^{7} - 40119 x^{6} + 11733 x^{5} + 69219 x^{4} + 7603 x^{3} - 9583 x^{2} - 13475 x + 2107 \)

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:  \(55507379263954413799411093504=2^{12}\cdot 7^{10}\cdot 13^{2}\cdot 883^{2}\cdot 19081^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.47$
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:  $2, 7, 13, 883, 19081$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} - \frac{1}{16} a^{3} + \frac{1}{16} a^{2} - \frac{7}{16} a + \frac{7}{16}$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{16} a^{2} - \frac{3}{8} a - \frac{5}{32}$, $\frac{1}{128} a^{9} + \frac{1}{128} a^{8} + \frac{1}{64} a^{7} + \frac{3}{64} a^{6} - \frac{1}{32} a^{5} - \frac{1}{16} a^{4} + \frac{11}{64} a^{3} - \frac{15}{64} a^{2} - \frac{21}{128} a + \frac{31}{128}$, $\frac{1}{256} a^{10} + \frac{1}{256} a^{8} + \frac{1}{64} a^{7} - \frac{5}{128} a^{6} + \frac{7}{64} a^{5} - \frac{1}{128} a^{4} + \frac{3}{64} a^{3} - \frac{55}{256} a^{2} + \frac{21}{64} a - \frac{63}{256}$, $\frac{1}{1024} a^{11} - \frac{1}{1024} a^{10} + \frac{1}{1024} a^{9} + \frac{3}{1024} a^{8} + \frac{9}{512} a^{7} + \frac{3}{512} a^{6} - \frac{63}{512} a^{5} + \frac{55}{512} a^{4} - \frac{99}{1024} a^{3} - \frac{85}{1024} a^{2} - \frac{51}{1024} a + \frac{223}{1024}$, $\frac{1}{2048} a^{12} + \frac{1}{512} a^{9} + \frac{21}{2048} a^{8} + \frac{3}{256} a^{7} - \frac{15}{256} a^{6} - \frac{1}{128} a^{5} + \frac{11}{2048} a^{4} - \frac{23}{256} a^{3} - \frac{17}{256} a^{2} + \frac{43}{512} a + \frac{223}{2048}$, $\frac{1}{4096} a^{13} - \frac{1}{4096} a^{12} + \frac{1}{1024} a^{10} - \frac{15}{4096} a^{9} - \frac{29}{4096} a^{8} + \frac{3}{256} a^{7} + \frac{21}{512} a^{6} + \frac{411}{4096} a^{5} + \frac{317}{4096} a^{4} + \frac{7}{256} a^{3} - \frac{3}{1024} a^{2} - \frac{557}{4096} a - \frac{447}{4096}$, $\frac{1}{11676871786496} a^{14} + \frac{186060939}{5838435893248} a^{13} - \frac{96970207}{11676871786496} a^{12} - \frac{901810059}{2919217946624} a^{11} + \frac{12275129757}{11676871786496} a^{10} - \frac{17709063523}{5838435893248} a^{9} + \frac{33054561725}{11676871786496} a^{8} + \frac{27425986795}{1459608973312} a^{7} + \frac{140466865683}{11676871786496} a^{6} + \frac{415273954661}{5838435893248} a^{5} - \frac{1258983040909}{11676871786496} a^{4} - \frac{287694884107}{2919217946624} a^{3} - \frac{1246680768241}{11676871786496} a^{2} - \frac{35460138827}{834062270464} a + \frac{820532394969}{1668124540928}$

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:  $8$
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:  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:  \( 480573521.927 \) (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

\(\Q(\zeta_{7})^+\)

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ $15$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
883Data not computed
19081Data not computed