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Properties

Label 16.0.87029511568...4601.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{6}\cdot 53^{12}$
Root discriminant $48.28$
Ramified primes $11, 53$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2333, 4675, -2741, 9972, 7276, -50833, 39317, 4513, -15357, 4547, 651, -401, 57, -38, 27, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 27*x^14 - 38*x^13 + 57*x^12 - 401*x^11 + 651*x^10 + 4547*x^9 - 15357*x^8 + 4513*x^7 + 39317*x^6 - 50833*x^5 + 7276*x^4 + 9972*x^3 - 2741*x^2 + 4675*x + 2333)
 
gp: K = bnfinit(x^16 - 8*x^15 + 27*x^14 - 38*x^13 + 57*x^12 - 401*x^11 + 651*x^10 + 4547*x^9 - 15357*x^8 + 4513*x^7 + 39317*x^6 - 50833*x^5 + 7276*x^4 + 9972*x^3 - 2741*x^2 + 4675*x + 2333, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 27 x^{14} - 38 x^{13} + 57 x^{12} - 401 x^{11} + 651 x^{10} + 4547 x^{9} - 15357 x^{8} + 4513 x^{7} + 39317 x^{6} - 50833 x^{5} + 7276 x^{4} + 9972 x^{3} - 2741 x^{2} + 4675 x + 2333 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(870295115683950043241964601=11^{6}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.28$
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:  $11, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{3927} a^{14} - \frac{166}{3927} a^{13} + \frac{185}{1309} a^{12} - \frac{326}{1309} a^{11} - \frac{10}{119} a^{10} + \frac{1366}{3927} a^{9} - \frac{1634}{3927} a^{8} - \frac{893}{3927} a^{7} - \frac{904}{3927} a^{6} + \frac{1636}{3927} a^{5} - \frac{835}{3927} a^{4} - \frac{106}{561} a^{3} + \frac{14}{561} a^{2} + \frac{1669}{3927} a - \frac{239}{3927}$, $\frac{1}{30611674460882657730208291391721} a^{15} + \frac{555223952985785766235422361}{10203891486960885910069430463907} a^{14} + \frac{1486913468082458156192691742934}{30611674460882657730208291391721} a^{13} + \frac{1546157801583525807877747660210}{10203891486960885910069430463907} a^{12} - \frac{4316903810011311944075414290977}{10203891486960885910069430463907} a^{11} + \frac{4663466269147663220923603523437}{30611674460882657730208291391721} a^{10} + \frac{71126278403579642743763802352}{214067653572605998113344695047} a^{9} - \frac{546409082715319344235971620692}{30611674460882657730208291391721} a^{8} + \frac{3567854599665828415971608090654}{10203891486960885910069430463907} a^{7} - \frac{883909198624180071628851058408}{10203891486960885910069430463907} a^{6} + \frac{21316170507937878957633692077}{351858327136582272761014843583} a^{5} + \frac{69740083783837289437875326509}{336392027042666568463827377931} a^{4} - \frac{4387329235659241817893440228389}{30611674460882657730208291391721} a^{3} + \frac{2767415631834526285968345255813}{10203891486960885910069430463907} a^{2} - \frac{6487799493414112439179193068033}{30611674460882657730208291391721} a - \frac{13589436123447598414180337457368}{30611674460882657730208291391721}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  $7$
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:  \( 1596248.2912 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.2.30899.1, 4.0.148877.1, 4.2.1637647.1, 8.2.29500764662699.1, 8.2.10502230211.1, 8.0.2681887696609.1

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$