↑

Properties

Label 16.0.76699264189...1216.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 17^{8}$
Root discriminant $23.32$
Ramified primes $2, 17$
Class number $4$
Class group $[2, 2]$
Galois group $C_2\times C_2^2.D_4$ (as 16T92)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 32, 32, -400, 1156, -1104, 480, 144, -60, 40, 8, -36, 1, -4, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^13 + x^12 - 36*x^11 + 8*x^10 + 40*x^9 - 60*x^8 + 144*x^7 + 480*x^6 - 1104*x^5 + 1156*x^4 - 400*x^3 + 32*x^2 + 32*x + 16)
 
gp: K = bnfinit(x^16 - 4*x^13 + x^12 - 36*x^11 + 8*x^10 + 40*x^9 - 60*x^8 + 144*x^7 + 480*x^6 - 1104*x^5 + 1156*x^4 - 400*x^3 + 32*x^2 + 32*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{13} + x^{12} - 36 x^{11} + 8 x^{10} + 40 x^{9} - 60 x^{8} + 144 x^{7} + 480 x^{6} - 1104 x^{5} + 1156 x^{4} - 400 x^{3} + 32 x^{2} + 32 x + 16 \)

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:  \(7669926418924454281216=2^{40}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.32$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{11} + \frac{1}{8} a^{9} - \frac{1}{16} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{12} - \frac{1}{4} a^{9} + \frac{3}{16} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{48} a^{13} - \frac{1}{48} a^{12} + \frac{1}{48} a^{11} - \frac{7}{48} a^{9} + \frac{3}{16} a^{8} - \frac{13}{48} a^{7} - \frac{1}{2} a^{5} - \frac{7}{24} a^{4} + \frac{7}{24} a^{3} - \frac{1}{6} a^{2} - \frac{1}{12} a + \frac{1}{3}$, $\frac{1}{425568} a^{14} + \frac{343}{212784} a^{13} + \frac{2125}{106392} a^{12} - \frac{477}{35464} a^{11} + \frac{9161}{425568} a^{10} - \frac{365}{70928} a^{9} + \frac{1273}{53196} a^{8} + \frac{162}{341} a^{7} - \frac{16967}{35464} a^{6} - \frac{2567}{26598} a^{5} - \frac{311}{9672} a^{4} - \frac{12137}{26598} a^{3} - \frac{6925}{106392} a^{2} + \frac{19373}{53196} a + \frac{5919}{17732}$, $\frac{1}{18201117792} a^{15} - \frac{205}{379189954} a^{14} + \frac{969863}{146783208} a^{13} - \frac{271242497}{9100558896} a^{12} - \frac{40555301}{18201117792} a^{11} - \frac{131138071}{4550279448} a^{10} + \frac{202048151}{1516759816} a^{9} - \frac{1950666149}{9100558896} a^{8} - \frac{1469263795}{9100558896} a^{7} - \frac{103331962}{568784931} a^{6} - \frac{6123224}{189594977} a^{5} - \frac{51979237}{239488392} a^{4} - \frac{78120557}{206830884} a^{3} + \frac{208969351}{568784931} a^{2} + \frac{182156968}{568784931} a + \frac{93998747}{1137569862}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{1466729}{827323536} a^{15} - \frac{33491537}{6067039264} a^{14} - \frac{3623613}{3033519632} a^{13} - \frac{38629075}{4550279448} a^{12} + \frac{185351651}{9100558896} a^{11} - \frac{1251120155}{18201117792} a^{10} + \frac{1936563599}{9100558896} a^{9} + \frac{40974053}{568784931} a^{8} - \frac{456355491}{1516759816} a^{7} + \frac{2797107821}{4550279448} a^{6} + \frac{116622269}{758379908} a^{5} - \frac{34631011}{7257224} a^{4} + \frac{1449435893}{189594977} a^{3} - \frac{9695172431}{1516759816} a^{2} + \frac{2620825525}{2275139724} a + \frac{610884499}{758379908} \) (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:  \( 179309.222586 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2.D_4$ (as 16T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2\times C_2^2.D_4$
Character table for $C_2\times C_2^2.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(i, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(i, \sqrt{34})\), \(\Q(\sqrt{-2}, \sqrt{-17})\), \(\Q(\zeta_{8})\), 8.4.10947264512.1, 8.0.5473632256.1, 8.4.43789058048.4

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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]$
2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_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}$