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Properties

Label 15.3.37214132212...9673.1
Degree $15$
Signature $[3, 6]$
Discriminant $3^{22}\cdot 17^{9}$
Root discriminant $27.42$
Ramified primes $3, 17$
Class number $1$
Class group Trivial
Galois group 15T41

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, -240, -720, -1096, -906, -459, -216, -81, 0, 4, 12, 9, -6, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 9*x^13 - 6*x^12 + 9*x^11 + 12*x^10 + 4*x^9 - 81*x^7 - 216*x^6 - 459*x^5 - 906*x^4 - 1096*x^3 - 720*x^2 - 240*x - 32)
 
gp: K = bnfinit(x^15 - 9*x^13 - 6*x^12 + 9*x^11 + 12*x^10 + 4*x^9 - 81*x^7 - 216*x^6 - 459*x^5 - 906*x^4 - 1096*x^3 - 720*x^2 - 240*x - 32, 1)
 

Normalized defining polynomial

\( x^{15} - 9 x^{13} - 6 x^{12} + 9 x^{11} + 12 x^{10} + 4 x^{9} - 81 x^{7} - 216 x^{6} - 459 x^{5} - 906 x^{4} - 1096 x^{3} - 720 x^{2} - 240 x - 32 \)

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:  \(3721413221257087109673=3^{22}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.42$
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:  $3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{15}{64} a^{10} - \frac{1}{16} a^{9} + \frac{13}{64} a^{8} - \frac{3}{32} a^{7} - \frac{3}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{64} a^{4} + \frac{5}{32} a^{3} + \frac{13}{64} a^{2} + \frac{7}{16} a + \frac{5}{16}$, $\frac{1}{512} a^{13} - \frac{1}{128} a^{12} + \frac{19}{512} a^{11} + \frac{63}{256} a^{10} - \frac{11}{512} a^{9} + \frac{1}{16} a^{8} - \frac{1}{4} a^{6} + \frac{175}{512} a^{5} - \frac{37}{128} a^{4} - \frac{71}{512} a^{3} + \frac{81}{256} a^{2} - \frac{9}{128} a - \frac{21}{64}$, $\frac{1}{4096} a^{14} + \frac{1}{2048} a^{13} - \frac{5}{4096} a^{12} - \frac{1}{256} a^{11} - \frac{23}{4096} a^{10} - \frac{17}{2048} a^{9} - \frac{1}{64} a^{8} - \frac{1}{32} a^{7} - \frac{337}{4096} a^{6} - \frac{445}{2048} a^{5} + \frac{1857}{4096} a^{4} - \frac{161}{512} a^{3} + \frac{53}{512} a^{2} + \frac{1}{32} a + \frac{1}{256}$

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:  \( 87526.6943784 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T41:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1620
The 21 conjugacy class representatives for [3^4]F(5)
Character table for [3^4]F(5) is not computed

Intermediate fields

5.1.44217.1

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

Sibling fields

Degree 15 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$3$3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.12.18.53$x^{12} - 12 x^{11} + 6 x^{10} + 12 x^{9} - 9 x^{8} + 3 x^{6} + 9 x^{5} - 9 x^{4} - 9 x^{3} - 9$$6$$2$$18$12T131$[3/2, 3/2, 2, 2]_{2}^{2}$
$17$17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.12.9.2$x^{12} - 34 x^{8} + 289 x^{4} - 44217$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$