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Properties

Label 16.0.12136551081...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $56.92$
Ramified primes $2, 5, 41$
Class number $3056$ (GRH)
Class group $[2, 2, 2, 382]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T268)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4536281, -2010692, 2075448, -1475320, 2287980, -1441988, 997404, -453580, 204159, -69300, 22796, -5716, 1440, -240, 52, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 52*x^14 - 240*x^13 + 1440*x^12 - 5716*x^11 + 22796*x^10 - 69300*x^9 + 204159*x^8 - 453580*x^7 + 997404*x^6 - 1441988*x^5 + 2287980*x^4 - 1475320*x^3 + 2075448*x^2 - 2010692*x + 4536281)
 
gp: K = bnfinit(x^16 - 4*x^15 + 52*x^14 - 240*x^13 + 1440*x^12 - 5716*x^11 + 22796*x^10 - 69300*x^9 + 204159*x^8 - 453580*x^7 + 997404*x^6 - 1441988*x^5 + 2287980*x^4 - 1475320*x^3 + 2075448*x^2 - 2010692*x + 4536281, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 52 x^{14} - 240 x^{13} + 1440 x^{12} - 5716 x^{11} + 22796 x^{10} - 69300 x^{9} + 204159 x^{8} - 453580 x^{7} + 997404 x^{6} - 1441988 x^{5} + 2287980 x^{4} - 1475320 x^{3} + 2075448 x^{2} - 2010692 x + 4536281 \)

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:  \(12136551081312256000000000000=2^{44}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.92$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{231} a^{14} + \frac{101}{231} a^{13} - \frac{25}{77} a^{12} - \frac{10}{21} a^{11} - \frac{27}{77} a^{10} - \frac{20}{231} a^{9} - \frac{5}{21} a^{8} + \frac{41}{231} a^{7} - \frac{74}{231} a^{6} - \frac{64}{231} a^{4} - \frac{107}{231} a^{3} + \frac{94}{231} a^{2} + \frac{37}{231} a + \frac{71}{231}$, $\frac{1}{16235942263811612957285665780309881697211976951} a^{15} - \frac{18577841996058096442560969043725551210727586}{16235942263811612957285665780309881697211976951} a^{14} - \frac{2435048861796597215731433733147195759322359108}{5411980754603870985761888593436627232403992317} a^{13} - \frac{693938192997291104126823153652814563852760834}{2319420323401658993897952254329983099601710993} a^{12} + \frac{1234727218640878745125570045578099664957047476}{5411980754603870985761888593436627232403992317} a^{11} - \frac{2814193095032706686528614799550845415215892959}{16235942263811612957285665780309881697211976951} a^{10} + \frac{787170666129843547271959791895886800794773051}{2319420323401658993897952254329983099601710993} a^{9} + \frac{107697105028396631401626034256471254708057504}{395998591800283242860625994641704431639316511} a^{8} + \frac{1064482353409016397112733679678222466592118172}{16235942263811612957285665780309881697211976951} a^{7} + \frac{2006349745164235412401561088160338348939823514}{5411980754603870985761888593436627232403992317} a^{6} - \frac{19436116570195085625408262224374835211658323}{16235942263811612957285665780309881697211976951} a^{5} + \frac{5520470943072041135912405334489576784963589899}{16235942263811612957285665780309881697211976951} a^{4} - \frac{1886895777337516192613613412359887817574167020}{16235942263811612957285665780309881697211976951} a^{3} - \frac{1509533132508749642636344736754153935366916569}{16235942263811612957285665780309881697211976951} a^{2} - \frac{490638201024942418646422863530134069246483436}{1475994751255601177935060525482716517928361541} a + \frac{50316315534523020138036285812405603095865232}{131999530600094414286875331547234810546438837}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{382}$, which has order $3056$ (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:  \( 7114.13535725 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T268):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed