Properties

Label 15.3.30638157055...2784.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{14}\cdot 3^{18}\cdot 13^{6}$
Root discriminant $19.91$
Ramified primes $2, 3, 13$
Class number $1$
Class group Trivial
Galois group $A_6$ (as 15T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 15, 0, -67, 30, -162, 68, -54, 0, 50, 12, 0, -2, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 2*x^12 + 12*x^10 + 50*x^9 - 54*x^7 + 68*x^6 - 162*x^5 + 30*x^4 - 67*x^3 + 15*x + 4)
 
gp: K = bnfinit(x^15 - 3*x^13 - 2*x^12 + 12*x^10 + 50*x^9 - 54*x^7 + 68*x^6 - 162*x^5 + 30*x^4 - 67*x^3 + 15*x + 4, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{13} - 2 x^{12} + 12 x^{10} + 50 x^{9} - 54 x^{7} + 68 x^{6} - 162 x^{5} + 30 x^{4} - 67 x^{3} + 15 x + 4 \)

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:  \(30638157055420022784=2^{14}\cdot 3^{18}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.91$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{2}{9} a^{5} - \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a^{2} + \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{11} - \frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{18} a^{6} - \frac{7}{18} a^{5} + \frac{5}{18} a^{4} - \frac{7}{18} a^{3} + \frac{5}{18} a^{2} - \frac{7}{18} a - \frac{1}{9}$, $\frac{1}{702} a^{13} - \frac{8}{351} a^{12} - \frac{1}{117} a^{11} - \frac{2}{27} a^{10} + \frac{23}{351} a^{9} + \frac{19}{39} a^{8} + \frac{146}{351} a^{7} + \frac{58}{351} a^{6} - \frac{53}{117} a^{5} + \frac{4}{39} a^{4} + \frac{14}{117} a^{3} - \frac{17}{39} a^{2} - \frac{79}{702} a + \frac{128}{351}$, $\frac{1}{5980356252} a^{14} - \frac{646705}{5980356252} a^{13} + \frac{21766499}{996726042} a^{12} + \frac{37570363}{2990178126} a^{11} + \frac{245957080}{1495089063} a^{10} - \frac{78032023}{996726042} a^{9} + \frac{394532939}{2990178126} a^{8} - \frac{195086908}{1495089063} a^{7} + \frac{7957133}{498363021} a^{6} + \frac{103565231}{996726042} a^{5} - \frac{290159579}{996726042} a^{4} - \frac{10850479}{996726042} a^{3} - \frac{1168436515}{5980356252} a^{2} + \frac{98787985}{460027404} a + \frac{238537894}{498363021}$

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

Class group and class number

Trivial group, which has 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:  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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 31837.7190213 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_6$ (as 15T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 7 conjugacy class representatives for $A_6$
Character table for $A_6$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 6 siblings: 6.2.7884864.1, 6.2.1971216.1
Degree 10 sibling: 10.2.15542770074624.1
Degree 15 sibling: Deg 15
Degree 20 sibling: 20.4.241577701592627342528741376.1
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: data not computed
Degree 45 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 R ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.6.6.7$x^{6} + 2 x^{2} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$3.6.6.1$x^{6} + 3 x^{5} - 2$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
3.9.12.21$x^{9} + 3 x^{4} + 6$$9$$1$$12$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$