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Properties

Label 16.0.52554919692...6256.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{38}\cdot 17^{6}\cdot 89^{2}$
Root discriminant $26.31$
Ramified primes $2, 17, 89$
Class number $2$
Class group $[2]$
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3464, -10768, 8332, 2332, -15, -8076, 5212, 120, -615, -328, 388, -20, -15, -12, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - 12*x^13 - 15*x^12 - 20*x^11 + 388*x^10 - 328*x^9 - 615*x^8 + 120*x^7 + 5212*x^6 - 8076*x^5 - 15*x^4 + 2332*x^3 + 8332*x^2 - 10768*x + 3464)
 
gp: K = bnfinit(x^16 + 4*x^14 - 12*x^13 - 15*x^12 - 20*x^11 + 388*x^10 - 328*x^9 - 615*x^8 + 120*x^7 + 5212*x^6 - 8076*x^5 - 15*x^4 + 2332*x^3 + 8332*x^2 - 10768*x + 3464, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} - 12 x^{13} - 15 x^{12} - 20 x^{11} + 388 x^{10} - 328 x^{9} - 615 x^{8} + 120 x^{7} + 5212 x^{6} - 8076 x^{5} - 15 x^{4} + 2332 x^{3} + 8332 x^{2} - 10768 x + 3464 \)

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:  \(52554919692301559136256=2^{38}\cdot 17^{6}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.31$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{14} a^{14} - \frac{1}{14} a^{13} + \frac{1}{7} a^{12} + \frac{3}{14} a^{11} - \frac{3}{14} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{2} a^{6} - \frac{3}{14} a^{5} - \frac{3}{14} a^{3} - \frac{5}{14} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{79212761526714750074709212} a^{15} + \frac{507586020995403185307243}{39606380763357375037354606} a^{14} - \frac{2084475576766430264462127}{39606380763357375037354606} a^{13} - \frac{1170816958685658421558556}{19803190381678687518677303} a^{12} - \frac{3103786670445532250417023}{79212761526714750074709212} a^{11} - \frac{8215028508359939792264539}{39606380763357375037354606} a^{10} - \frac{9308397219825751103479684}{19803190381678687518677303} a^{9} + \frac{954415374206774471389010}{2829027197382669645525329} a^{8} + \frac{12451595637104474456134069}{79212761526714750074709212} a^{7} + \frac{11988031741560515814269805}{39606380763357375037354606} a^{6} - \frac{419237751842691026732547}{921078622403659884589642} a^{5} + \frac{4056713797014865957977574}{19803190381678687518677303} a^{4} - \frac{25818635769299960140394707}{79212761526714750074709212} a^{3} - \frac{5697464160110579819436143}{39606380763357375037354606} a^{2} + \frac{1204834165415112437138500}{2829027197382669645525329} a + \frac{8659191189985105912035550}{19803190381678687518677303}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{2165436225998271719060013}{79212761526714750074709212} a^{15} + \frac{491125781738609193379935}{19803190381678687518677303} a^{14} + \frac{5231121036609595233946047}{39606380763357375037354606} a^{13} - \frac{4112960726287421638456713}{19803190381678687518677303} a^{12} - \frac{47288820567865716107316651}{79212761526714750074709212} a^{11} - \frac{43114542231173909112832833}{39606380763357375037354606} a^{10} + \frac{190353998504497026119369311}{19803190381678687518677303} a^{9} - \frac{5230451987092393655442642}{19803190381678687518677303} a^{8} - \frac{192166413734834431826616153}{11316108789530678582101316} a^{7} - \frac{238088258253727408639853554}{19803190381678687518677303} a^{6} + \frac{17298540216050596276285221}{131582660343379983512806} a^{5} - \frac{2016829989028278020105339084}{19803190381678687518677303} a^{4} - \frac{7274051662809780092053875783}{79212761526714750074709212} a^{3} - \frac{749214699961564977341926061}{39606380763357375037354606} a^{2} + \frac{4144949844241078252259296864}{19803190381678687518677303} a - \frac{300749195269351396449573687}{2829027197382669645525329} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 599635.796811 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), 4.4.4352.1, 4.0.1088.2, \(\Q(\zeta_{8})\), 8.0.18939904.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.22.87$x^{8} + 4 x^{7} + 6 x^{4} + 12 x^{2} + 2$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$17$17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$