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Properties

Label 16.12.7416612542...2688.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{36}\cdot 3^{5}\cdot 13^{8}\cdot 859^{4}$
Root discriminant $130.88$
Ramified primes $2, 3, 13, 859$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1581

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![559872, 0, -1479168, 0, -628416, 0, 616848, 0, -125103, 0, 7576, 0, 254, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 + 254*x^12 + 7576*x^10 - 125103*x^8 + 616848*x^6 - 628416*x^4 - 1479168*x^2 + 559872)
 
gp: K = bnfinit(x^16 - 40*x^14 + 254*x^12 + 7576*x^10 - 125103*x^8 + 616848*x^6 - 628416*x^4 - 1479168*x^2 + 559872, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} + 254 x^{12} + 7576 x^{10} - 125103 x^{8} + 616848 x^{6} - 628416 x^{4} - 1479168 x^{2} + 559872 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7416612542850992781854363445362688=2^{36}\cdot 3^{5}\cdot 13^{8}\cdot 859^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.88$
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:  $2, 3, 13, 859$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{24} a^{8} - \frac{1}{24} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{48} a^{9} - \frac{1}{48} a^{8} - \frac{1}{48} a^{5} + \frac{1}{48} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{144} a^{10} - \frac{1}{48} a^{8} + \frac{11}{144} a^{6} - \frac{1}{16} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{288} a^{11} - \frac{1}{288} a^{10} - \frac{1}{96} a^{9} + \frac{1}{96} a^{8} + \frac{11}{288} a^{7} - \frac{11}{288} a^{6} + \frac{7}{32} a^{5} - \frac{7}{32} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{288} a^{12} + \frac{1}{144} a^{8} + \frac{1}{12} a^{6} - \frac{17}{96} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{5184} a^{13} + \frac{1}{2592} a^{11} - \frac{1}{288} a^{10} + \frac{1}{162} a^{9} + \frac{1}{96} a^{8} + \frac{59}{2592} a^{7} - \frac{11}{288} a^{6} - \frac{131}{1728} a^{5} - \frac{7}{32} a^{4} - \frac{1}{27} a^{3} + \frac{1}{3} a^{2} - \frac{13}{36} a - \frac{1}{2}$, $\frac{1}{68088092890752} a^{14} + \frac{632731607}{4255505805672} a^{12} - \frac{71628595805}{34044046445376} a^{10} - \frac{1189979773}{386864164152} a^{8} - \frac{167220951367}{2063275542144} a^{6} - \frac{9642292301}{709250967612} a^{4} + \frac{43185352373}{472833978408} a^{2} + \frac{1653622016}{6567138589}$, $\frac{1}{408528557344512} a^{15} + \frac{632731607}{25533034834032} a^{13} + \frac{164788393399}{204264278672256} a^{11} + \frac{45978060973}{4642369969824} a^{9} - \frac{1}{48} a^{8} - \frac{525428510767}{12379653252864} a^{7} - \frac{1870800835535}{17022023222688} a^{5} + \frac{1}{48} a^{4} - \frac{626662783705}{2837003870448} a^{3} - \frac{1}{4} a^{2} - \frac{1913442527}{6567138589} a$

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:  $13$
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:  \( 6684926549250 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1581:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1581 are not computed
Character table for t16n1581 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.435513.1, 4.4.32448.1, 4.4.9290944.2, 8.8.776894763700224.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.20.42$x^{8} + 16 x^{7} + 8 x^{6} + 240$$4$$2$$20$$(C_4^2 : C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
2.8.16.15$x^{8} + 4 x^{6} + 4 x^{5} + 2 x^{4} + 12 x^{2} + 8 x + 28$$4$$2$$16$$C_2^2:C_4$$[2, 2, 3]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
859Data not computed