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Properties

Label 16.0.57456715021...1504.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 7^{2}\cdot 17^{10}\cdot 23^{2}$
Root discriminant $62.73$
Ramified primes $2, 7, 17, 23$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T860

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29964676, 0, -27917400, 0, 11464256, 0, -2678996, 0, 381955, 0, -33880, 0, 1900, 0, -64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 64*x^14 + 1900*x^12 - 33880*x^10 + 381955*x^8 - 2678996*x^6 + 11464256*x^4 - 27917400*x^2 + 29964676)
 
gp: K = bnfinit(x^16 - 64*x^14 + 1900*x^12 - 33880*x^10 + 381955*x^8 - 2678996*x^6 + 11464256*x^4 - 27917400*x^2 + 29964676, 1)
 

Normalized defining polynomial

\( x^{16} - 64 x^{14} + 1900 x^{12} - 33880 x^{10} + 381955 x^{8} - 2678996 x^{6} + 11464256 x^{4} - 27917400 x^{2} + 29964676 \)

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:  \(57456715021727885253362581504=2^{40}\cdot 7^{2}\cdot 17^{10}\cdot 23^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.73$
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:  $2, 7, 17, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{340} a^{12} + \frac{19}{170} a^{10} - \frac{18}{85} a^{8} - \frac{59}{170} a^{6} - \frac{7}{68} a^{4} + \frac{1}{5} a^{2} + \frac{3}{10}$, $\frac{1}{340} a^{13} + \frac{19}{170} a^{11} - \frac{18}{85} a^{9} - \frac{59}{170} a^{7} - \frac{7}{68} a^{5} + \frac{1}{5} a^{3} + \frac{3}{10} a$, $\frac{1}{26698029695454749660} a^{14} + \frac{6474361860326274}{6674507423863687415} a^{12} - \frac{283429533051758507}{6674507423863687415} a^{10} - \frac{113458686540088338}{953501060551955345} a^{8} + \frac{1188856160709026823}{3814004242207821380} a^{6} + \frac{2125854323169994862}{6674507423863687415} a^{4} + \frac{329713082289505769}{785236167513374990} a^{2} - \frac{822786306093018}{2438621638240295}$, $\frac{1}{53396059390909499320} a^{15} - \frac{1}{53396059390909499320} a^{14} - \frac{52626169310032403}{53396059390909499320} a^{13} + \frac{52626169310032403}{53396059390909499320} a^{12} + \frac{511378371021165685}{10679211878181899864} a^{11} - \frac{511378371021165685}{10679211878181899864} a^{10} - \frac{599664320127122993}{7628008484415642760} a^{9} + \frac{599664320127122993}{7628008484415642760} a^{8} + \frac{389759731348475901}{1907002121103910690} a^{7} - \frac{389759731348475901}{1907002121103910690} a^{6} - \frac{2192944598153567583}{13349014847727374830} a^{5} + \frac{2192944598153567583}{13349014847727374830} a^{4} + \frac{282641966271759133}{785236167513374990} a^{3} - \frac{282641966271759133}{785236167513374990} a^{2} - \frac{335061978444959}{4877243276480590} a + \frac{335061978444959}{4877243276480590}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 129470793.567 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T860:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n860 are not computed
Character table for t16n860 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.0.1088.2 x2, 4.0.2312.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.11.16$x^{4} + 14$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.10.3$x^{4} + 6 x^{2} - 9$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$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}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$