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Properties

Label 16.12.6767652375...1904.2
Degree $16$
Signature $[12, 2]$
Discriminant $2^{62}\cdot 1889^{5}\cdot 247007^{2}$
Root discriminant $731.82$
Ramified primes $2, 1889, 247007$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T1186

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1645032138861793448324, 0, -100599134011721249248, 0, 400452722366707216, 0, 3209410207612384, 0, -13391862498668, 0, 16501892048, 0, -6229112, 0, -1392, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1392*x^14 - 6229112*x^12 + 16501892048*x^10 - 13391862498668*x^8 + 3209410207612384*x^6 + 400452722366707216*x^4 - 100599134011721249248*x^2 + 1645032138861793448324)
 
gp: K = bnfinit(x^16 - 1392*x^14 - 6229112*x^12 + 16501892048*x^10 - 13391862498668*x^8 + 3209410207612384*x^6 + 400452722366707216*x^4 - 100599134011721249248*x^2 + 1645032138861793448324, 1)
 

Normalized defining polynomial

\( x^{16} - 1392 x^{14} - 6229112 x^{12} + 16501892048 x^{10} - 13391862498668 x^{8} + 3209410207612384 x^{6} + 400452722366707216 x^{4} - 100599134011721249248 x^{2} + 1645032138861793448324 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6767652375800250507037432977960083710388731904=2^{62}\cdot 1889^{5}\cdot 247007^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $731.82$
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:  $2, 1889, 247007$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{16} a^{8} - \frac{1}{4} a^{4} + \frac{1}{8}$, $\frac{1}{16} a^{9} - \frac{1}{4} a^{5} + \frac{1}{8} a$, $\frac{1}{16} a^{10} - \frac{1}{4} a^{6} + \frac{1}{8} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{4} a^{7} + \frac{1}{8} a^{3}$, $\frac{1}{1057840} a^{12} + \frac{4633}{211568} a^{10} - \frac{6409}{264460} a^{8} - \frac{59541}{264460} a^{6} + \frac{26847}{75560} a^{4} - \frac{204627}{528920} a^{2} - \frac{16}{35}$, $\frac{1}{1057840} a^{13} + \frac{4633}{211568} a^{11} - \frac{6409}{264460} a^{9} - \frac{59541}{264460} a^{7} + \frac{26847}{75560} a^{5} - \frac{204627}{528920} a^{3} - \frac{16}{35} a$, $\frac{1}{21498544878603560384172608850316523177448355049822892777822169044179360} a^{14} + \frac{13405562509136194139451914727639109776796162687031162314640711}{38390258711792072114593944375565219959729205446112308531825301864606} a^{12} + \frac{54183503352358169565727840468416771539316310827837078493161974339253}{2687318109825445048021576106289565397181044381227861597227771130522420} a^{10} - \frac{68227462043869801982825500885480571288473691975341666942513244932951}{5374636219650890096043152212579130794362088762455723194455542261044840} a^{8} - \frac{79380943493682924192264808765870866001337950790509828832447374997351}{10749272439301780192086304425158261588724177524911446388911084522089680} a^{6} - \frac{6899704999127425264269307627244371862961472241557531766137715921989}{1343659054912722524010788053144782698590522190613930798613885565261210} a^{4} - \frac{224552454461456100415720359866235214910924236768305435417252671337}{1422614139664078903134767658173406774579695278574834090644664441780} a^{2} + \frac{29659270134592804375045480820103930218823902092267669191}{609783821445155119969656358822864613963964846999987006172}$, $\frac{1}{21498544878603560384172608850316523177448355049822892777822169044179360} a^{15} + \frac{13405562509136194139451914727639109776796162687031162314640711}{38390258711792072114593944375565219959729205446112308531825301864606} a^{13} + \frac{54183503352358169565727840468416771539316310827837078493161974339253}{2687318109825445048021576106289565397181044381227861597227771130522420} a^{11} - \frac{68227462043869801982825500885480571288473691975341666942513244932951}{5374636219650890096043152212579130794362088762455723194455542261044840} a^{9} - \frac{79380943493682924192264808765870866001337950790509828832447374997351}{10749272439301780192086304425158261588724177524911446388911084522089680} a^{7} - \frac{6899704999127425264269307627244371862961472241557531766137715921989}{1343659054912722524010788053144782698590522190613930798613885565261210} a^{5} - \frac{224552454461456100415720359866235214910924236768305435417252671337}{1422614139664078903134767658173406774579695278574834090644664441780} a^{3} + \frac{29659270134592804375045480820103930218823902092267669191}{609783821445155119969656358822864613963964846999987006172} a$

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:  $13$
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:  \( 76340361025200000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1186:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 46 conjugacy class representatives for t16n1186
Character table for t16n1186 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

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 $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.31.36$x^{8} + 12 x^{4} + 2$$8$$1$$31$$(C_8:C_2):C_2$$[2, 3, 7/2, 4, 5]$
2.8.31.36$x^{8} + 12 x^{4} + 2$$8$$1$$31$$(C_8:C_2):C_2$$[2, 3, 7/2, 4, 5]$
1889Data not computed
247007Data not computed