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Properties

Label 16.0.46375012149...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{10}\cdot 11^{2}\cdot 29^{4}\cdot 71^{2}$
Root discriminant $71.47$
Ramified primes $2, 3, 5, 11, 29, 71$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8581, -19342, 207530, -103868, 70601, -79736, 67554, -33434, 25201, -11174, 6910, -2232, 891, -198, 50, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 50*x^14 - 198*x^13 + 891*x^12 - 2232*x^11 + 6910*x^10 - 11174*x^9 + 25201*x^8 - 33434*x^7 + 67554*x^6 - 79736*x^5 + 70601*x^4 - 103868*x^3 + 207530*x^2 - 19342*x + 8581)
 
gp: K = bnfinit(x^16 - 6*x^15 + 50*x^14 - 198*x^13 + 891*x^12 - 2232*x^11 + 6910*x^10 - 11174*x^9 + 25201*x^8 - 33434*x^7 + 67554*x^6 - 79736*x^5 + 70601*x^4 - 103868*x^3 + 207530*x^2 - 19342*x + 8581, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 50 x^{14} - 198 x^{13} + 891 x^{12} - 2232 x^{11} + 6910 x^{10} - 11174 x^{9} + 25201 x^{8} - 33434 x^{7} + 67554 x^{6} - 79736 x^{5} + 70601 x^{4} - 103868 x^{3} + 207530 x^{2} - 19342 x + 8581 \)

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:  \(463750121497395363840000000000=2^{24}\cdot 3^{8}\cdot 5^{10}\cdot 11^{2}\cdot 29^{4}\cdot 71^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.47$
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:  $2, 3, 5, 11, 29, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{3}{7} a^{10} - \frac{2}{7} a^{9} - \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{1435018852868121520558488276176363859479} a^{15} - \frac{97968574779230797307006522774835864492}{1435018852868121520558488276176363859479} a^{14} + \frac{12260108392075811195984142100838009103}{205002693266874502936926896596623408497} a^{13} + \frac{68934156522713171068994248796987208936}{205002693266874502936926896596623408497} a^{12} - \frac{57146850934749761807547328817340843729}{1435018852868121520558488276176363859479} a^{11} - \frac{286761191085028919583692474643430352839}{1435018852868121520558488276176363859479} a^{10} + \frac{59183875101782689731992152734183420072}{130456259351647410959862570561487623589} a^{9} - \frac{512380117753162054416238183109757788811}{1435018852868121520558488276176363859479} a^{8} + \frac{517073163399304201146763627589771874962}{1435018852868121520558488276176363859479} a^{7} - \frac{654983149555868631577594817891708076535}{1435018852868121520558488276176363859479} a^{6} + \frac{152808980818448270731129086703357908982}{1435018852868121520558488276176363859479} a^{5} + \frac{495558905778358778455481204999418913238}{1435018852868121520558488276176363859479} a^{4} - \frac{456885199427115076947145728092152357091}{1435018852868121520558488276176363859479} a^{3} - \frac{437623831857089274128497393892780453666}{1435018852868121520558488276176363859479} a^{2} - \frac{35606858625525921488751188681698090132}{205002693266874502936926896596623408497} a - \frac{188905241267401286006196558677250757028}{1435018852868121520558488276176363859479}$

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

Class group and class number

$C_{2}\times C_{4}$, 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:  \( 16100444.7881 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}, \sqrt{5})\), 4.0.11600.1, 4.0.6525.1, 8.0.10899360000.14

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$