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Properties

Label 16.0.67108864000...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{14}$
Root discriminant $23.13$
Ramified primes $2, 5$
Class number $2$
Class group $[2]$
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![241, -660, 584, -480, 872, -980, 632, -560, 545, -260, 268, -40, 102, 0, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 + 102*x^12 - 40*x^11 + 268*x^10 - 260*x^9 + 545*x^8 - 560*x^7 + 632*x^6 - 980*x^5 + 872*x^4 - 480*x^3 + 584*x^2 - 660*x + 241)
 
gp: K = bnfinit(x^16 + 16*x^14 + 102*x^12 - 40*x^11 + 268*x^10 - 260*x^9 + 545*x^8 - 560*x^7 + 632*x^6 - 980*x^5 + 872*x^4 - 480*x^3 + 584*x^2 - 660*x + 241, 1)
 

Normalized defining polynomial

\( x^{16} + 16 x^{14} + 102 x^{12} - 40 x^{11} + 268 x^{10} - 260 x^{9} + 545 x^{8} - 560 x^{7} + 632 x^{6} - 980 x^{5} + 872 x^{4} - 480 x^{3} + 584 x^{2} - 660 x + 241 \)

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:  \(6710886400000000000000=2^{40}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.13$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{55519699} a^{14} + \frac{11477974}{55519699} a^{13} - \frac{25660000}{55519699} a^{12} - \frac{18666076}{55519699} a^{11} - \frac{21699348}{55519699} a^{10} + \frac{13782898}{55519699} a^{9} - \frac{19729735}{55519699} a^{8} - \frac{23972310}{55519699} a^{7} - \frac{24665691}{55519699} a^{6} - \frac{20139199}{55519699} a^{5} - \frac{2068168}{55519699} a^{4} - \frac{4026812}{55519699} a^{3} + \frac{8012}{197579} a^{2} + \frac{6558618}{55519699} a + \frac{25847762}{55519699}$, $\frac{1}{561333082653179} a^{15} - \frac{202423}{561333082653179} a^{14} - \frac{112202439836100}{561333082653179} a^{13} + \frac{43306401854431}{561333082653179} a^{12} - \frac{5210353959708}{13691050796419} a^{11} + \frac{37444910093483}{561333082653179} a^{10} - \frac{199283342171511}{561333082653179} a^{9} - \frac{58391897145383}{561333082653179} a^{8} + \frac{158184860634977}{561333082653179} a^{7} - \frac{46438841330493}{561333082653179} a^{6} + \frac{32376183386431}{561333082653179} a^{5} + \frac{95646959693234}{561333082653179} a^{4} - \frac{139911776133753}{561333082653179} a^{3} - \frac{160643202237771}{561333082653179} a^{2} + \frac{221757347373085}{561333082653179} a + \frac{170280253327934}{561333082653179}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 7114.13535725 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{20})^+\), 4.4.8000.1, \(\Q(\zeta_{40})^+\), 8.0.5120000000.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed