↑

Properties

Label 16.8.14510321368...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 29^{8}\cdot 109^{2}$
Root discriminant $32.37$
Ramified primes $5, 29, 109$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2.C_2\wr C_2^2$ (as 16T400)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-499, -281, 2436, -747, -618, -1306, 42, 993, -144, 398, -148, -61, -58, -7, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^14 - 7*x^13 - 58*x^12 - 61*x^11 - 148*x^10 + 398*x^9 - 144*x^8 + 993*x^7 + 42*x^6 - 1306*x^5 - 618*x^4 - 747*x^3 + 2436*x^2 - 281*x - 499)
 
gp: K = bnfinit(x^16 - x^15 + x^14 - 7*x^13 - 58*x^12 - 61*x^11 - 148*x^10 + 398*x^9 - 144*x^8 + 993*x^7 + 42*x^6 - 1306*x^5 - 618*x^4 - 747*x^3 + 2436*x^2 - 281*x - 499, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + x^{14} - 7 x^{13} - 58 x^{12} - 61 x^{11} - 148 x^{10} + 398 x^{9} - 144 x^{8} + 993 x^{7} + 42 x^{6} - 1306 x^{5} - 618 x^{4} - 747 x^{3} + 2436 x^{2} - 281 x - 499 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1451032136813877197265625=5^{12}\cdot 29^{8}\cdot 109^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.37$
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:  $5, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{9} + \frac{1}{5} a^{6} - \frac{1}{5} a^{3} + \frac{1}{5}$, $\frac{1}{15} a^{13} - \frac{1}{15} a^{12} - \frac{1}{3} a^{11} + \frac{4}{15} a^{10} + \frac{2}{5} a^{9} - \frac{1}{3} a^{8} + \frac{1}{15} a^{7} - \frac{1}{15} a^{6} + \frac{1}{3} a^{5} - \frac{1}{15} a^{4} + \frac{1}{15} a^{3} + \frac{1}{15} a - \frac{1}{15}$, $\frac{1}{45} a^{14} + \frac{1}{15} a^{12} + \frac{14}{45} a^{11} - \frac{4}{9} a^{10} - \frac{8}{45} a^{9} + \frac{11}{45} a^{8} + \frac{1}{3} a^{7} - \frac{17}{45} a^{6} - \frac{11}{45} a^{5} + \frac{1}{3} a^{4} - \frac{8}{45} a^{3} + \frac{16}{45} a^{2} - \frac{7}{45}$, $\frac{1}{13679835119995169064825} a^{15} - \frac{59820864717063698207}{13679835119995169064825} a^{14} - \frac{95963572976475868184}{4559945039998389688275} a^{13} - \frac{27122587993794318970}{547193404799806762593} a^{12} + \frac{1239376261622716391854}{4559945039998389688275} a^{11} - \frac{1818379119046819186738}{13679835119995169064825} a^{10} + \frac{361287913036444786289}{911989007999677937655} a^{9} + \frac{3356999590226903030393}{13679835119995169064825} a^{8} - \frac{5878799304662405488772}{13679835119995169064825} a^{7} - \frac{1073585993284096288481}{2735967023999033812965} a^{6} + \frac{1755824265709694408767}{13679835119995169064825} a^{5} + \frac{4840474101923418945262}{13679835119995169064825} a^{4} - \frac{921386676558863273897}{2735967023999033812965} a^{3} - \frac{4331182143464294182157}{13679835119995169064825} a^{2} - \frac{2168954057804408855242}{13679835119995169064825} a - \frac{8112911626774018066}{27414499238467272675}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 463367.25398 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.C_2\wr C_2^2$ (as 16T400):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $C_2.C_2\wr C_2^2$
Character table for $C_2.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$