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Properties

Label 16.8.43149301773...5625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 101^{10}$
Root discriminant $40.01$
Ramified primes $5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-80, 520, -204, -2078, -501, 2649, 3018, -1586, -2244, 591, 890, -84, -380, 127, -10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 10*x^14 + 127*x^13 - 380*x^12 - 84*x^11 + 890*x^10 + 591*x^9 - 2244*x^8 - 1586*x^7 + 3018*x^6 + 2649*x^5 - 501*x^4 - 2078*x^3 - 204*x^2 + 520*x - 80)
 
gp: K = bnfinit(x^16 - 4*x^15 - 10*x^14 + 127*x^13 - 380*x^12 - 84*x^11 + 890*x^10 + 591*x^9 - 2244*x^8 - 1586*x^7 + 3018*x^6 + 2649*x^5 - 501*x^4 - 2078*x^3 - 204*x^2 + 520*x - 80, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 10 x^{14} + 127 x^{13} - 380 x^{12} - 84 x^{11} + 890 x^{10} + 591 x^{9} - 2244 x^{8} - 1586 x^{7} + 3018 x^{6} + 2649 x^{5} - 501 x^{4} - 2078 x^{3} - 204 x^{2} + 520 x - 80 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(43149301773875176172265625=5^{8}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.01$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{10} a^{8} + \frac{1}{10} a^{7} - \frac{3}{10} a^{6} + \frac{3}{10} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{10} a^{2}$, $\frac{1}{10} a^{9} + \frac{1}{10} a^{7} + \frac{1}{10} a^{6} + \frac{3}{10} a^{4} + \frac{3}{10} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{10} - \frac{1}{20} a^{8} - \frac{1}{20} a^{7} + \frac{3}{10} a^{6} - \frac{3}{20} a^{5} - \frac{3}{20} a^{4} - \frac{1}{20} a^{3} - \frac{3}{20} a^{2}$, $\frac{1}{100} a^{11} + \frac{1}{50} a^{10} + \frac{3}{100} a^{9} + \frac{3}{100} a^{8} + \frac{6}{25} a^{7} + \frac{9}{20} a^{6} + \frac{39}{100} a^{5} - \frac{37}{100} a^{4} + \frac{43}{100} a^{3} - \frac{2}{25} a^{2} - \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{100} a^{12} - \frac{1}{100} a^{10} - \frac{3}{100} a^{9} - \frac{1}{50} a^{8} - \frac{23}{100} a^{7} + \frac{9}{100} a^{6} + \frac{1}{4} a^{5} - \frac{43}{100} a^{4} - \frac{7}{50} a^{3} + \frac{3}{50} a^{2} + \frac{1}{5}$, $\frac{1}{1000} a^{13} - \frac{1}{1000} a^{12} + \frac{1}{250} a^{11} + \frac{1}{125} a^{10} + \frac{3}{500} a^{9} - \frac{3}{500} a^{8} + \frac{121}{500} a^{7} - \frac{69}{1000} a^{6} + \frac{327}{1000} a^{5} - \frac{193}{500} a^{4} + \frac{1}{200} a^{3} - \frac{32}{125} a^{2} + \frac{1}{50} a + \frac{12}{25}$, $\frac{1}{5000} a^{14} - \frac{1}{2500} a^{13} + \frac{1}{1000} a^{12} + \frac{3}{625} a^{11} - \frac{14}{625} a^{10} + \frac{37}{1250} a^{9} - \frac{121}{2500} a^{8} + \frac{219}{5000} a^{7} - \frac{113}{625} a^{6} - \frac{83}{5000} a^{5} + \frac{1301}{5000} a^{4} - \frac{151}{5000} a^{3} - \frac{817}{2500} a^{2} - \frac{57}{250} a - \frac{42}{125}$, $\frac{1}{7427991225475000} a^{15} - \frac{120130897589}{3713995612737500} a^{14} - \frac{678573252093}{7427991225475000} a^{13} - \frac{3705943374989}{1856997806368750} a^{12} - \frac{1084382995617}{928498903184375} a^{11} + \frac{13651493851121}{742799122547500} a^{10} + \frac{14634723379113}{371399561273750} a^{9} - \frac{226818551606789}{7427991225475000} a^{8} - \frac{606860798504949}{3713995612737500} a^{7} - \frac{773516039812879}{7427991225475000} a^{6} + \frac{196762202949809}{7427991225475000} a^{5} + \frac{2167609428395823}{7427991225475000} a^{4} + \frac{287714326333499}{928498903184375} a^{3} - \frac{55284628272157}{928498903184375} a^{2} - \frac{20258335610901}{185699780636875} a - \frac{40860712894583}{185699780636875}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 35504723.7489 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{505}) \), 4.4.51005.1 x2, 4.4.2525.1 x2, \(\Q(\sqrt{5}, \sqrt{101})\), 8.4.6568812813125.2 x2, 8.4.65037750625.1 x2, 8.8.65037750625.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
101Data not computed