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Properties

Label 16.0.65534439047...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{8}\cdot 5^{8}\cdot 61^{2}$
Root discriminant $30.80$
Ramified primes $2, 3, 5, 61$
Class number $2$
Class group $[2]$
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![850, -2920, 4956, -8880, 14382, -15508, 11524, -5288, 2779, -1000, 506, -88, 72, -12, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 10*x^14 - 12*x^13 + 72*x^12 - 88*x^11 + 506*x^10 - 1000*x^9 + 2779*x^8 - 5288*x^7 + 11524*x^6 - 15508*x^5 + 14382*x^4 - 8880*x^3 + 4956*x^2 - 2920*x + 850)
 
gp: K = bnfinit(x^16 + 10*x^14 - 12*x^13 + 72*x^12 - 88*x^11 + 506*x^10 - 1000*x^9 + 2779*x^8 - 5288*x^7 + 11524*x^6 - 15508*x^5 + 14382*x^4 - 8880*x^3 + 4956*x^2 - 2920*x + 850, 1)
 

Normalized defining polynomial

\( x^{16} + 10 x^{14} - 12 x^{13} + 72 x^{12} - 88 x^{11} + 506 x^{10} - 1000 x^{9} + 2779 x^{8} - 5288 x^{7} + 11524 x^{6} - 15508 x^{5} + 14382 x^{4} - 8880 x^{3} + 4956 x^{2} - 2920 x + 850 \)

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:  \(655344390478233600000000=2^{36}\cdot 3^{8}\cdot 5^{8}\cdot 61^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.80$
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:  $2, 3, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9}$, $\frac{1}{440} a^{14} - \frac{27}{110} a^{13} + \frac{109}{440} a^{12} - \frac{1}{10} a^{11} - \frac{1}{440} a^{10} - \frac{5}{11} a^{9} - \frac{19}{40} a^{8} - \frac{17}{110} a^{7} + \frac{21}{55} a^{6} + \frac{7}{110} a^{5} - \frac{9}{22} a^{4} - \frac{1}{55} a^{3} - \frac{47}{220} a^{2} + \frac{9}{55} a + \frac{13}{44}$, $\frac{1}{2172146996624093819210320} a^{15} + \frac{92920168608428366683}{434429399324818763842064} a^{14} + \frac{107712104635016436050373}{434429399324818763842064} a^{13} - \frac{350483254617928662373397}{2172146996624093819210320} a^{12} - \frac{26275044821908148900613}{2172146996624093819210320} a^{11} + \frac{809713931687272286257437}{2172146996624093819210320} a^{10} + \frac{44566614650914496415911}{2172146996624093819210320} a^{9} - \frac{71329253416834893910459}{434429399324818763842064} a^{8} - \frac{45243323552765462594049}{543036749156023454802580} a^{7} + \frac{102442098238410242754073}{543036749156023454802580} a^{6} + \frac{18066521858101242593789}{135759187289005863700645} a^{5} + \frac{185472958574020158791153}{543036749156023454802580} a^{4} + \frac{405939277502587785013281}{1086073498312046909605160} a^{3} + \frac{6129800538594554098449}{19746790878400852901912} a^{2} - \frac{418218913852628545475267}{1086073498312046909605160} a + \frac{11647962444353920181235}{217214699662409381921032}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{119788537041}{49967330377844} a^{15} + \frac{46116318815}{24983665188922} a^{14} + \frac{1263418204131}{49967330377844} a^{13} - \frac{118640050495}{12491832594461} a^{12} + \frac{8208870941351}{49967330377844} a^{11} - \frac{2186708026971}{24983665188922} a^{10} + \frac{56938545161385}{49967330377844} a^{9} - \frac{19210783838019}{12491832594461} a^{8} + \frac{68030731080716}{12491832594461} a^{7} - \frac{107074506126755}{12491832594461} a^{6} + \frac{262134426176125}{12491832594461} a^{5} - \frac{269045826937537}{12491832594461} a^{4} + \frac{468311321344133}{24983665188922} a^{3} - \frac{120363134394001}{12491832594461} a^{2} + \frac{169913853013115}{24983665188922} a - \frac{39202645697323}{12491832594461} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1614321.58086 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), 4.0.1280.1, 4.0.11520.1, \(\Q(i, \sqrt{15})\), 8.0.3317760000.11

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.8.6$x^{4} + 6 x^{2} + 4 x + 2$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
2.4.8.6$x^{4} + 6 x^{2} + 4 x + 2$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
2.8.20.75$x^{8} + 8 x^{6} + 16 x^{5} + 80$$8$$1$$20$$C_2^3 : C_4 $$[2, 3, 3]^{2}$
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$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$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61Data not computed