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Properties

Label 16.0.22659849813...0784.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{68}\cdot 449^{4}\cdot 1889$
Root discriminant $140.35$
Ramified primes $2, 449, 1889$
Class number $7128$ (GRH)
Class group $[2, 2, 1782]$ (GRH)
Galois group 16T1604

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![155307329, -145993456, 177167392, -64732256, 80597868, -14153040, 17993888, -1881184, 2242654, -148464, 160000, -5824, 6332, -80, 128, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 128*x^14 - 80*x^13 + 6332*x^12 - 5824*x^11 + 160000*x^10 - 148464*x^9 + 2242654*x^8 - 1881184*x^7 + 17993888*x^6 - 14153040*x^5 + 80597868*x^4 - 64732256*x^3 + 177167392*x^2 - 145993456*x + 155307329)
 
gp: K = bnfinit(x^16 + 128*x^14 - 80*x^13 + 6332*x^12 - 5824*x^11 + 160000*x^10 - 148464*x^9 + 2242654*x^8 - 1881184*x^7 + 17993888*x^6 - 14153040*x^5 + 80597868*x^4 - 64732256*x^3 + 177167392*x^2 - 145993456*x + 155307329, 1)
 

Normalized defining polynomial

\( x^{16} + 128 x^{14} - 80 x^{13} + 6332 x^{12} - 5824 x^{11} + 160000 x^{10} - 148464 x^{9} + 2242654 x^{8} - 1881184 x^{7} + 17993888 x^{6} - 14153040 x^{5} + 80597868 x^{4} - 64732256 x^{3} + 177167392 x^{2} - 145993456 x + 155307329 \)

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:  \(22659849813268057575626593255030784=2^{68}\cdot 449^{4}\cdot 1889\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $140.35$
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:  $2, 449, 1889$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{3251452} a^{14} - \frac{244477}{3251452} a^{13} + \frac{46553}{1625726} a^{12} - \frac{19634}{812863} a^{11} - \frac{752413}{3251452} a^{10} - \frac{805941}{3251452} a^{9} + \frac{267695}{1625726} a^{8} - \frac{222703}{1625726} a^{7} + \frac{515493}{3251452} a^{6} - \frac{107957}{3251452} a^{5} - \frac{227780}{812863} a^{4} + \frac{481431}{1625726} a^{3} + \frac{544715}{3251452} a^{2} - \frac{1069037}{3251452} a - \frac{12170}{812863}$, $\frac{1}{267644371855166376881860545968527237510846468} a^{15} + \frac{17769295073040802309514108972289391365}{267644371855166376881860545968527237510846468} a^{14} + \frac{3459630660845075597211255501197049365257925}{133822185927583188440930272984263618755423234} a^{13} + \frac{211627191959576237329266224763421418648362}{2158422353670696587756939886842961592829407} a^{12} + \frac{2789965947765428058946581386651788279303024}{66911092963791594220465136492131809377711617} a^{11} - \frac{13765311747259721293139880250514616179376139}{133822185927583188440930272984263618755423234} a^{10} + \frac{19794417528324503981356259668732068348050359}{267644371855166376881860545968527237510846468} a^{9} - \frac{8312825735404188069488586609637500631097347}{267644371855166376881860545968527237510846468} a^{8} + \frac{349672646371462327176279854948828269515805}{8633689414682786351027759547371846371317628} a^{7} + \frac{35041763064102062625073768140365938115293083}{267644371855166376881860545968527237510846468} a^{6} - \frac{59026457464356447727624270502911383023913143}{133822185927583188440930272984263618755423234} a^{5} + \frac{6669555466481111109112300775232127156826948}{66911092963791594220465136492131809377711617} a^{4} + \frac{9076353580745432794379572010980176270525510}{66911092963791594220465136492131809377711617} a^{3} - \frac{17960864024914052631111356619534783561889431}{133822185927583188440930272984263618755423234} a^{2} + \frac{6946093110556341201056989432105687789129425}{267644371855166376881860545968527237510846468} a - \frac{120441596308799424068899003498639701702264917}{267644371855166376881860545968527237510846468}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1782}$, which has order $7128$ (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:  \( 8221413.04148 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1604:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1604 are not computed
Character table for t16n1604 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.432934850920448.2

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 $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
449Data not computed
1889Data not computed