Properties

Label 15.3.18953525353...7584.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{26}\cdot 3^{24}$
Root discriminant $19.28$
Ramified primes $2, 3$
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![-9, 27, -81, 63, -18, -90, 108, 0, -54, 34, -12, -12, 17, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 3*x^13 + 17*x^12 - 12*x^11 - 12*x^10 + 34*x^9 - 54*x^8 + 108*x^6 - 90*x^5 - 18*x^4 + 63*x^3 - 81*x^2 + 27*x - 9)
 
gp: K = bnfinit(x^15 - 3*x^14 - 3*x^13 + 17*x^12 - 12*x^11 - 12*x^10 + 34*x^9 - 54*x^8 + 108*x^6 - 90*x^5 - 18*x^4 + 63*x^3 - 81*x^2 + 27*x - 9, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 3 x^{13} + 17 x^{12} - 12 x^{11} - 12 x^{10} + 34 x^{9} - 54 x^{8} + 108 x^{6} - 90 x^{5} - 18 x^{4} + 63 x^{3} - 81 x^{2} + 27 x - 9 \)

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:  \(18953525353286467584=2^{26}\cdot 3^{24}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.28$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6}$, $\frac{1}{15} a^{10} + \frac{1}{15} a^{8} - \frac{2}{15} a^{7} + \frac{1}{5} a^{6} + \frac{4}{15} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{30} a^{11} - \frac{1}{30} a^{10} + \frac{1}{30} a^{9} - \frac{1}{10} a^{8} + \frac{1}{6} a^{7} + \frac{1}{30} a^{6} + \frac{1}{6} a^{5} - \frac{1}{10} a^{4} - \frac{1}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{30} a^{12} - \frac{1}{15} a^{9} + \frac{1}{15} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{15} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{10}$, $\frac{1}{30} a^{13} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} + \frac{1}{15} a^{7} + \frac{4}{15} a^{6} - \frac{4}{15} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{217020} a^{14} + \frac{34}{54255} a^{13} - \frac{2801}{217020} a^{12} + \frac{219}{36170} a^{11} - \frac{545}{21702} a^{10} - \frac{9847}{108510} a^{9} + \frac{2872}{54255} a^{8} - \frac{2717}{7234} a^{7} - \frac{241}{36170} a^{6} + \frac{15301}{108510} a^{5} + \frac{2411}{18085} a^{4} - \frac{9741}{36170} a^{3} - \frac{31397}{72340} a^{2} + \frac{9797}{36170} a - \frac{18111}{72340}$

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:  \( 25215.0957342 \)
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: 6.2.1679616.2, 6.2.6718464.1
Degree 10 sibling: 10.2.11284439629824.1
Degree 15 sibling: Deg 15
Degree 20 sibling: 20.4.127338577759142414150270976.1
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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.6.10.1$x^{6} + 2 x^{5} + 2 x^{4} + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
2.8.16.9$x^{8} + 2 x^{4} + 8 x + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
$3$3.6.8.1$x^{6} + 6 x^{5} + 18 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$
3.9.16.4$x^{9} + 3 x^{8} + 3$$9$$1$$16$$C_3^2:C_4$$[2, 2]^{4}$