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Properties

Label 15.11.3128917033...6281.1
Degree $15$
Signature $[11, 2]$
Discriminant $3^{22}\cdot 59^{2}\cdot 83^{2}\cdot 401^{6}$
Root discriminant $171.01$
Ramified primes $3, 59, 83, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T80

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1410336, -10577520, 31732560, -47335848, 34121178, -7159347, -3552616, 1341405, -15012, 6330, 3528, -2646, -6, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 9*x^13 - 6*x^12 - 2646*x^11 + 3528*x^10 + 6330*x^9 - 15012*x^8 + 1341405*x^7 - 3552616*x^6 - 7159347*x^5 + 34121178*x^4 - 47335848*x^3 + 31732560*x^2 - 10577520*x + 1410336)
 
gp: K = bnfinit(x^15 + 9*x^13 - 6*x^12 - 2646*x^11 + 3528*x^10 + 6330*x^9 - 15012*x^8 + 1341405*x^7 - 3552616*x^6 - 7159347*x^5 + 34121178*x^4 - 47335848*x^3 + 31732560*x^2 - 10577520*x + 1410336, 1)
 

Normalized defining polynomial

\( x^{15} + 9 x^{13} - 6 x^{12} - 2646 x^{11} + 3528 x^{10} + 6330 x^{9} - 15012 x^{8} + 1341405 x^{7} - 3552616 x^{6} - 7159347 x^{5} + 34121178 x^{4} - 47335848 x^{3} + 31732560 x^{2} - 10577520 x + 1410336 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3128917033950916372142938576036281=3^{22}\cdot 59^{2}\cdot 83^{2}\cdot 401^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $171.01$
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:  $3, 59, 83, 401$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{16} a^{5} + \frac{1}{8} a^{2} + \frac{7}{16} a + \frac{3}{8}$, $\frac{1}{96} a^{6} - \frac{1}{32} a^{5} + \frac{1}{48} a^{3} - \frac{5}{32} a^{2} + \frac{11}{32} a - \frac{3}{16}$, $\frac{1}{192} a^{7} - \frac{1}{192} a^{6} - \frac{1}{32} a^{5} - \frac{5}{96} a^{4} + \frac{1}{192} a^{3} - \frac{3}{64} a^{2} + \frac{3}{16} a + \frac{7}{16}$, $\frac{1}{384} a^{8} + \frac{1}{384} a^{6} + \frac{1}{48} a^{5} - \frac{3}{128} a^{4} + \frac{1}{48} a^{3} + \frac{1}{128} a^{2} + \frac{3}{8} a + \frac{3}{32}$, $\frac{1}{2304} a^{9} - \frac{1}{768} a^{8} - \frac{1}{768} a^{7} - \frac{11}{2304} a^{6} + \frac{1}{768} a^{5} - \frac{23}{768} a^{4} + \frac{79}{2304} a^{3} - \frac{17}{256} a^{2} - \frac{1}{16} a - \frac{71}{192}$, $\frac{1}{4608} a^{10} + \frac{1}{1152} a^{7} - \frac{1}{256} a^{6} + \frac{3}{128} a^{5} - \frac{47}{1152} a^{4} - \frac{1}{128} a^{3} - \frac{31}{512} a^{2} - \frac{17}{384} a - \frac{47}{128}$, $\frac{1}{9216} a^{11} - \frac{1}{9216} a^{10} + \frac{1}{2304} a^{8} - \frac{11}{4608} a^{7} + \frac{5}{1536} a^{6} - \frac{1}{1152} a^{5} + \frac{19}{1152} a^{4} - \frac{145}{3072} a^{3} + \frac{505}{3072} a^{2} + \frac{95}{192} a + \frac{95}{256}$, $\frac{1}{9216} a^{12} - \frac{1}{9216} a^{10} - \frac{1}{1536} a^{8} + \frac{5}{2304} a^{7} - \frac{5}{1536} a^{6} - \frac{13}{768} a^{5} - \frac{7}{9216} a^{4} + \frac{143}{2304} a^{3} - \frac{249}{1024} a^{2} + \frac{113}{768} a - \frac{343}{768}$, $\frac{1}{294912} a^{13} - \frac{1}{73728} a^{12} - \frac{11}{294912} a^{11} + \frac{11}{147456} a^{10} - \frac{1}{147456} a^{9} - \frac{1}{36864} a^{8} + \frac{17}{147456} a^{7} + \frac{205}{73728} a^{6} + \frac{1285}{294912} a^{5} + \frac{239}{24576} a^{4} - \frac{20791}{294912} a^{3} + \frac{545}{49152} a^{2} + \frac{143}{8192} a + \frac{2227}{12288}$, $\frac{1}{9437184} a^{14} - \frac{1}{4718592} a^{13} + \frac{13}{9437184} a^{12} - \frac{1}{294912} a^{11} - \frac{89}{1572864} a^{10} + \frac{125}{2359296} a^{9} - \frac{5527}{4718592} a^{8} - \frac{129}{131072} a^{7} + \frac{28781}{9437184} a^{6} - \frac{2315}{1572864} a^{5} + \frac{249425}{9437184} a^{4} - \frac{20185}{1179648} a^{3} + \frac{3797}{131072} a^{2} + \frac{1117}{24576} a + \frac{81763}{196608}$

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:  $12$
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:  \( 571892290212 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T80:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38880
The 48 conjugacy class representatives for [1/2.S(3)^5]D(5)
Character table for [1/2.S(3)^5]D(5) is not computed

Intermediate fields

5.5.160801.1

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

Sibling fields

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 $15$ $15$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ R

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$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.6.10.4$x^{6} + 3 x^{3} + 72$$3$$2$$10$$S_3^2$$[3/2, 5/2]_{2}^{2}$
3.6.8.1$x^{6} + 6 x^{5} + 18 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$
59Data not computed
83Data not computed
401Data not computed