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Properties

Label 16.0.56118626684...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{46}\cdot 3^{12}\cdot 5^{4}\cdot 7^{4}$
Root discriminant $40.67$
Ramified primes $2, 3, 5, 7$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $C_2^4:C_2^2$ (as 16T119)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![376, -384, -528, 4928, -4264, -5904, 17520, -19200, 16512, -11872, 6792, -2952, 1044, -272, 60, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 60*x^14 - 272*x^13 + 1044*x^12 - 2952*x^11 + 6792*x^10 - 11872*x^9 + 16512*x^8 - 19200*x^7 + 17520*x^6 - 5904*x^5 - 4264*x^4 + 4928*x^3 - 528*x^2 - 384*x + 376)
 
gp: K = bnfinit(x^16 - 8*x^15 + 60*x^14 - 272*x^13 + 1044*x^12 - 2952*x^11 + 6792*x^10 - 11872*x^9 + 16512*x^8 - 19200*x^7 + 17520*x^6 - 5904*x^5 - 4264*x^4 + 4928*x^3 - 528*x^2 - 384*x + 376, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 60 x^{14} - 272 x^{13} + 1044 x^{12} - 2952 x^{11} + 6792 x^{10} - 11872 x^{9} + 16512 x^{8} - 19200 x^{7} + 17520 x^{6} - 5904 x^{5} - 4264 x^{4} + 4928 x^{3} - 528 x^{2} - 384 x + 376 \)

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:  \(56118626684141976944640000=2^{46}\cdot 3^{12}\cdot 5^{4}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.67$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{20} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{20} a^{13} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{2300} a^{14} - \frac{29}{2300} a^{13} - \frac{1}{115} a^{12} + \frac{1}{575} a^{11} + \frac{24}{115} a^{10} + \frac{41}{1150} a^{9} - \frac{43}{230} a^{8} - \frac{62}{575} a^{6} - \frac{48}{575} a^{5} + \frac{266}{575} a^{4} - \frac{19}{115} a^{3} + \frac{32}{115} a^{2} - \frac{268}{575} a - \frac{209}{575}$, $\frac{1}{12366871134036273323300} a^{15} - \frac{20270419988140601}{263124917745452623900} a^{14} + \frac{15839254113212674353}{3091717783509068330825} a^{13} - \frac{133140818718196855483}{6183435567018136661650} a^{12} - \frac{38297267822767042927}{12366871134036273323300} a^{11} + \frac{1524362984646721730481}{6183435567018136661650} a^{10} + \frac{39729335352176832299}{268845024652962463550} a^{9} - \frac{110596773308047923489}{618343556701813666165} a^{8} - \frac{1492332541365287749969}{6183435567018136661650} a^{7} - \frac{738697557622488416067}{3091717783509068330825} a^{6} + \frac{109981095198112743711}{618343556701813666165} a^{5} - \frac{226936903075426748208}{3091717783509068330825} a^{4} + \frac{119551226145088503304}{618343556701813666165} a^{3} - \frac{1073345745150761382073}{3091717783509068330825} a^{2} + \frac{301357793870492418437}{618343556701813666165} a + \frac{11851356214775392356}{65781229436363155975}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 454019.00739 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2$ (as 16T119):

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 $C_2^4:C_2^2$
Character table for $C_2^4:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), 4.0.27648.1 x2, 4.0.13824.1 x2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.26011238400.1, 8.0.3057647616.7, 8.8.3745618329600.1

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 R R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.0.1$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}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$