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Properties

Label 16.4.89815699851...4032.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 113^{6}\cdot 137^{3}$
Root discriminant $41.89$
Ramified primes $2, 113, 137$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1638

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1681, -17302, 43559, -69276, 89074, -91756, 67424, -30050, 4165, 3312, -2015, 410, -33, 14, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^13 - 33*x^12 + 410*x^11 - 2015*x^10 + 3312*x^9 + 4165*x^8 - 30050*x^7 + 67424*x^6 - 91756*x^5 + 89074*x^4 - 69276*x^3 + 43559*x^2 - 17302*x + 1681)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^13 - 33*x^12 + 410*x^11 - 2015*x^10 + 3312*x^9 + 4165*x^8 - 30050*x^7 + 67424*x^6 - 91756*x^5 + 89074*x^4 - 69276*x^3 + 43559*x^2 - 17302*x + 1681, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{13} - 33 x^{12} + 410 x^{11} - 2015 x^{10} + 3312 x^{9} + 4165 x^{8} - 30050 x^{7} + 67424 x^{6} - 91756 x^{5} + 89074 x^{4} - 69276 x^{3} + 43559 x^{2} - 17302 x + 1681 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(89815699851913524288684032=2^{24}\cdot 113^{6}\cdot 137^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.89$
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:  $2, 113, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{104880770520962215313651680082} a^{15} + \frac{1390042975230829973994076496}{52440385260481107656825840041} a^{14} - \frac{344767463170540450116318447}{2558067573682005251552480002} a^{13} - \frac{19135817238914163067055130945}{52440385260481107656825840041} a^{12} - \frac{2137719088541897359975712863}{52440385260481107656825840041} a^{11} - \frac{35328698974968063438729690}{75237281578882507398602353} a^{10} - \frac{4155419329945672919999792073}{104880770520962215313651680082} a^{9} - \frac{18554390577801511798173703980}{52440385260481107656825840041} a^{8} + \frac{2791795438874295321218718979}{52440385260481107656825840041} a^{7} - \frac{8755336299524647544391524015}{52440385260481107656825840041} a^{6} + \frac{10211407369313382333646845779}{52440385260481107656825840041} a^{5} - \frac{12899651219344391306800907775}{52440385260481107656825840041} a^{4} - \frac{21230383802668936886214034177}{52440385260481107656825840041} a^{3} - \frac{5549399646778391799392683654}{52440385260481107656825840041} a^{2} - \frac{2973242590730138380609879165}{8067751578535555024127052314} a + \frac{198571913554525195072126086}{1279033786841002625776240001}$

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:  $9$
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:  \( 6090640.96944 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1638:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1638 are not computed
Character table for t16n1638 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7232.1, 8.8.5910106112.1

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 $16$ $16$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$113$113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$137$137.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
137.2.1.2$x^{2} + 411$$2$$1$$1$$C_2$$[\ ]_{2}$
137.4.0.1$x^{4} - x + 26$$1$$4$$0$$C_4$$[\ ]^{4}$
137.4.0.1$x^{4} - x + 26$$1$$4$$0$$C_4$$[\ ]^{4}$
137.4.2.1$x^{4} + 1507 x^{2} + 675684$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$