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Properties

Label 16.4.52792485092...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{16}\cdot 5^{8}\cdot 11^{4}\cdot 269^{5}$
Root discriminant $46.79$
Ramified primes $2, 5, 11, 269$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1870

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4304, 0, -6080, 4544, 10956, -5152, 648, -3096, 708, 332, 122, -2, -44, -14, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 14*x^13 - 44*x^12 - 2*x^11 + 122*x^10 + 332*x^9 + 708*x^8 - 3096*x^7 + 648*x^6 - 5152*x^5 + 10956*x^4 + 4544*x^3 - 6080*x^2 - 4304)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 - 14*x^13 - 44*x^12 - 2*x^11 + 122*x^10 + 332*x^9 + 708*x^8 - 3096*x^7 + 648*x^6 - 5152*x^5 + 10956*x^4 + 4544*x^3 - 6080*x^2 - 4304, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} - 14 x^{13} - 44 x^{12} - 2 x^{11} + 122 x^{10} + 332 x^{9} + 708 x^{8} - 3096 x^{7} + 648 x^{6} - 5152 x^{5} + 10956 x^{4} + 4544 x^{3} - 6080 x^{2} - 4304 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(527924850922027750400000000=2^{16}\cdot 5^{8}\cdot 11^{4}\cdot 269^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.79$
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:  $2, 5, 11, 269$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{16} a^{14} + \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{15366818142587562992471922784} a^{15} + \frac{68653231646509841031837017}{3841704535646890748117980696} a^{14} + \frac{25733727189172547689133685}{3841704535646890748117980696} a^{13} + \frac{5598276272881082635629881}{48938911282125996791311856} a^{12} - \frac{415293717982547394968294007}{1920852267823445374058990348} a^{11} - \frac{1187908379488923078498942343}{7683409071293781496235961392} a^{10} - \frac{874768099499719377269402989}{7683409071293781496235961392} a^{9} - \frac{165401897731021452883770871}{3841704535646890748117980696} a^{8} - \frac{394643272205598075786230511}{1920852267823445374058990348} a^{7} - \frac{132985533282565394537102189}{960426133911722687029495174} a^{6} - \frac{97739520412468227822267693}{1920852267823445374058990348} a^{5} - \frac{19899609090656459815774253}{960426133911722687029495174} a^{4} - \frac{61741628463455522974486129}{3841704535646890748117980696} a^{3} + \frac{317107348936379472618585429}{1920852267823445374058990348} a^{2} + \frac{579933197692314554907277285}{1920852267823445374058990348} a - \frac{162738548993530316595961305}{480213066955861343514747587}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 3571279.01469 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1870:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 83 conjugacy class representatives for t16n1870 are not computed
Character table for t16n1870 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.87556810000.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.12.12.29$x^{12} + 6 x^{10} + 51 x^{8} - 252 x^{6} - 393 x^{4} - 234 x^{2} - 203$$6$$2$$12$12T159$[4/3, 4/3, 4/3, 4/3]_{3}^{12}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
269Data not computed