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Properties

Label 16.0.10589292162...0384.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 17^{8}\cdot 47^{2}$
Root discriminant $31.74$
Ramified primes $2, 17, 47$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T364)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -1280, 8704, -14336, 16420, 128, 800, -1280, 88, -112, 160, 0, 13, -24, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^13 + 13*x^12 + 160*x^10 - 112*x^9 + 88*x^8 - 1280*x^7 + 800*x^6 + 128*x^5 + 16420*x^4 - 14336*x^3 + 8704*x^2 - 1280*x + 64)
 
gp: K = bnfinit(x^16 - 24*x^13 + 13*x^12 + 160*x^10 - 112*x^9 + 88*x^8 - 1280*x^7 + 800*x^6 + 128*x^5 + 16420*x^4 - 14336*x^3 + 8704*x^2 - 1280*x + 64, 1)
 

Normalized defining polynomial

\( x^{16} - 24 x^{13} + 13 x^{12} + 160 x^{10} - 112 x^{9} + 88 x^{8} - 1280 x^{7} + 800 x^{6} + 128 x^{5} + 16420 x^{4} - 14336 x^{3} + 8704 x^{2} - 1280 x + 64 \)

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:  \(1058929216212757469200384=2^{36}\cdot 17^{8}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.74$
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:  $2, 17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{10} + \frac{3}{8} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{11} - \frac{5}{16} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{816} a^{12} - \frac{13}{816} a^{11} - \frac{3}{136} a^{10} + \frac{25}{204} a^{9} - \frac{133}{816} a^{8} + \frac{361}{816} a^{7} - \frac{55}{408} a^{6} - \frac{91}{204} a^{5} - \frac{179}{408} a^{4} + \frac{79}{408} a^{3} + \frac{19}{68} a^{2} + \frac{41}{102} a + \frac{1}{51}$, $\frac{1}{1632} a^{13} - \frac{1}{48} a^{11} - \frac{1}{51} a^{10} + \frac{49}{544} a^{9} + \frac{11}{68} a^{8} + \frac{277}{816} a^{7} - \frac{11}{68} a^{6} - \frac{403}{816} a^{5} - \frac{1}{204} a^{4} - \frac{169}{408} a^{3} + \frac{20}{51} a^{2} - \frac{13}{102} a - \frac{19}{51}$, $\frac{1}{11424} a^{14} + \frac{1}{5712} a^{13} + \frac{1}{1904} a^{12} - \frac{1}{1428} a^{11} - \frac{229}{11424} a^{10} + \frac{239}{5712} a^{9} + \frac{47}{816} a^{8} + \frac{127}{357} a^{7} - \frac{2255}{5712} a^{6} + \frac{647}{2856} a^{5} + \frac{381}{952} a^{4} - \frac{41}{102} a^{3} + \frac{20}{51} a^{2} + \frac{1}{17} a - \frac{40}{119}$, $\frac{1}{1725020409231819168} a^{15} + \frac{909433726659}{35937925192329566} a^{14} + \frac{299370150392953}{1725020409231819168} a^{13} + \frac{12691024108185}{287503401538636528} a^{12} - \frac{3637820233968625}{246431487033117024} a^{11} - \frac{1563002065837061}{25367947194585576} a^{10} + \frac{28109498625967987}{1725020409231819168} a^{9} + \frac{68368657451860313}{862510204615909584} a^{8} + \frac{9804338615607565}{22697636963576568} a^{7} + \frac{71469606492673805}{143751700769318264} a^{6} - \frac{111710759149753391}{862510204615909584} a^{5} + \frac{115975754091394537}{431255102307954792} a^{4} + \frac{18053102275401445}{61607871758279256} a^{3} + \frac{1925444931548189}{10267978626379876} a^{2} + \frac{7323560887665556}{17968962596164783} a - \frac{3099019414342756}{53906887788494349}$

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:  $7$
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:  \( 1166140.10872 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T364):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{2}) \), 4.0.1088.2 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 4.0.2312.1 x2, 8.4.1029042864128.2, 8.4.1029042864128.1, 8.0.342102016.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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]$
2.4.10.2$x^{4} + 2 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.10.2$x^{4} + 2 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$