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Properties

Label 16.0.10026747971...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{6}\cdot 1621^{4}$
Root discriminant $75.00$
Ramified primes $5, 29, 1621$
Class number $27624$ (GRH)
Class group $[2, 2, 6906]$ (GRH)
Galois group 16T1276

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![194676821, 203130409, 159770709, 100830379, 49007769, 19095936, 9604628, 1498347, 1258001, 19015, 101693, -3878, 4689, -205, 110, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 110*x^14 - 205*x^13 + 4689*x^12 - 3878*x^11 + 101693*x^10 + 19015*x^9 + 1258001*x^8 + 1498347*x^7 + 9604628*x^6 + 19095936*x^5 + 49007769*x^4 + 100830379*x^3 + 159770709*x^2 + 203130409*x + 194676821)
 
gp: K = bnfinit(x^16 - 3*x^15 + 110*x^14 - 205*x^13 + 4689*x^12 - 3878*x^11 + 101693*x^10 + 19015*x^9 + 1258001*x^8 + 1498347*x^7 + 9604628*x^6 + 19095936*x^5 + 49007769*x^4 + 100830379*x^3 + 159770709*x^2 + 203130409*x + 194676821, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 110 x^{14} - 205 x^{13} + 4689 x^{12} - 3878 x^{11} + 101693 x^{10} + 19015 x^{9} + 1258001 x^{8} + 1498347 x^{7} + 9604628 x^{6} + 19095936 x^{5} + 49007769 x^{4} + 100830379 x^{3} + 159770709 x^{2} + 203130409 x + 194676821 \)

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:  \(1002674797153076232861572265625=5^{12}\cdot 29^{6}\cdot 1621^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.00$
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:  $5, 29, 1621$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{83} a^{14} - \frac{1}{83} a^{13} + \frac{15}{83} a^{12} + \frac{1}{83} a^{11} - \frac{24}{83} a^{10} - \frac{35}{83} a^{9} + \frac{22}{83} a^{8} - \frac{13}{83} a^{7} - \frac{27}{83} a^{6} + \frac{24}{83} a^{5} + \frac{20}{83} a^{4} + \frac{9}{83} a^{3} - \frac{12}{83} a^{2} + \frac{39}{83} a - \frac{26}{83}$, $\frac{1}{1038209248747415623004431960565462216724325955978117} a^{15} - \frac{232914558316538347548315428227704285537174799129}{1038209248747415623004431960565462216724325955978117} a^{14} - \frac{403893927178505756491302477601489929241903157037641}{1038209248747415623004431960565462216724325955978117} a^{13} - \frac{501007389863373610366478508013852836204226445034065}{1038209248747415623004431960565462216724325955978117} a^{12} + \frac{15259270508102509238318287305057319773062738912431}{1038209248747415623004431960565462216724325955978117} a^{11} + \frac{238921286456025392654288594256809614043261653412500}{1038209248747415623004431960565462216724325955978117} a^{10} + \frac{145312410920740444903716681936162678174141745702632}{1038209248747415623004431960565462216724325955978117} a^{9} + \frac{484220008631017480641955878689655785135919738466178}{1038209248747415623004431960565462216724325955978117} a^{8} - \frac{49209892374876460338477264552419925254122580698821}{1038209248747415623004431960565462216724325955978117} a^{7} - \frac{179616631560382659660551400187635001284363310751763}{1038209248747415623004431960565462216724325955978117} a^{6} + \frac{36613823164605439040388455906870350536510247197559}{1038209248747415623004431960565462216724325955978117} a^{5} - \frac{430124966108220055472604139557445885913883109779116}{1038209248747415623004431960565462216724325955978117} a^{4} - \frac{384332021257610332339677430521819755002475103377016}{1038209248747415623004431960565462216724325955978117} a^{3} - \frac{260773630246722901503270499563384355938047526705352}{1038209248747415623004431960565462216724325955978117} a^{2} + \frac{170228752412800786122709745030270222869017320721656}{1038209248747415623004431960565462216724325955978117} a - \frac{244375696032530589529510269985457892727626959244332}{1038209248747415623004431960565462216724325955978117}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6906}$, which has order $27624$ (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:  \( 5728.02182166 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1276:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1276
Character table for t16n1276 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.852038125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
1621Data not computed