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Properties

Label 16.0.55691650270...681.25
Degree $16$
Signature $[0, 8]$
Discriminant $41^{14}\cdot 59^{8}$
Root discriminant $197.98$
Ramified primes $41, 59$
Class number $1728$ (GRH)
Class group $[12, 12, 12]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16429086976, 0, -4660210752, 0, 647985536, 0, -4143076, 0, 388601, 0, -35236, 0, 366, 0, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 + 366*x^12 - 35236*x^10 + 388601*x^8 - 4143076*x^6 + 647985536*x^4 - 4660210752*x^2 + 16429086976)
 
gp: K = bnfinit(x^16 - 32*x^14 + 366*x^12 - 35236*x^10 + 388601*x^8 - 4143076*x^6 + 647985536*x^4 - 4660210752*x^2 + 16429086976, 1)
 

Normalized defining polynomial

\( x^{16} - 32 x^{14} + 366 x^{12} - 35236 x^{10} + 388601 x^{8} - 4143076 x^{6} + 647985536 x^{4} - 4660210752 x^{2} + 16429086976 \)

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:  \(5569165027023164184679061585237737681=41^{14}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $197.98$
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:  $41, 59$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{5} + \frac{1}{16} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} + \frac{1}{32} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{9} + \frac{1}{32} a^{5} - \frac{1}{16} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{64} a^{10} - \frac{1}{64} a^{9} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} + \frac{3}{64} a^{5} + \frac{3}{32} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{9} - \frac{1}{64} a^{7} + \frac{1}{64} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{2}$, $\frac{1}{7552} a^{12} - \frac{37}{7552} a^{10} - \frac{1}{64} a^{9} - \frac{1}{7552} a^{8} - \frac{1}{32} a^{7} - \frac{411}{7552} a^{6} + \frac{3}{64} a^{5} - \frac{5}{472} a^{4} + \frac{9}{472} a^{2} - \frac{1}{2} a + \frac{35}{118}$, $\frac{1}{15104} a^{13} - \frac{1}{15104} a^{12} + \frac{81}{15104} a^{11} - \frac{81}{15104} a^{10} + \frac{235}{15104} a^{9} - \frac{235}{15104} a^{8} + \frac{179}{15104} a^{7} - \frac{179}{15104} a^{6} + \frac{113}{944} a^{5} - \frac{113}{944} a^{4} + \frac{17}{236} a^{3} - \frac{17}{236} a^{2} - \frac{83}{236} a + \frac{83}{236}$, $\frac{1}{7436417299903328237551616} a^{14} - \frac{75037307179038072667}{1859104324975832059387904} a^{12} - \frac{22978596577143509842625}{3718208649951664118775808} a^{10} + \frac{25787158431613438209021}{1859104324975832059387904} a^{8} - \frac{41854824717838043676535}{7436417299903328237551616} a^{6} - \frac{1}{8} a^{5} - \frac{25090128531281533286505}{464776081243958014846976} a^{4} - \frac{1}{8} a^{3} - \frac{4068155315342351858831}{116194020310989503711744} a^{2} + \frac{1}{4} a - \frac{34373603031644493642557}{116194020310989503711744}$, $\frac{1}{238292555958102250044103983104} a^{15} - \frac{1}{14872834599806656475103232} a^{14} - \frac{825003346760835451077019}{59573138989525562511025995776} a^{13} - \frac{171136464666278672485}{3718208649951664118775808} a^{12} + \frac{196329652955140708124805311}{119146277979051125022051991552} a^{11} - \frac{16901554461797802871999}{7436417299903328237551616} a^{10} + \frac{685814465201168394231944253}{59573138989525562511025995776} a^{9} - \frac{25540984659768121463869}{3718208649951664118775808} a^{8} + \frac{3418932728498089964330069385}{238292555958102250044103983104} a^{7} - \frac{599181677167366760699273}{14872834599806656475103232} a^{6} - \frac{594518255492257584853441321}{14893284747381390627756498944} a^{5} - \frac{71656163803927947558231}{929552162487916029693952} a^{4} - \frac{170463157689255397464438799}{3723321186845347656939124736} a^{3} - \frac{12671661170139186811505}{232388040621979007423488} a^{2} + \frac{1417539567056651969658498499}{3723321186845347656939124736} a - \frac{90725026699850678723}{232388040621979007423488}$

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

Class group and class number

$C_{12}\times C_{12}\times C_{12}$, which has order $1728$ (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:  \( 1833256639.05 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

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 $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{-59}) \), \(\Q(\sqrt{-2419}) \), 4.0.239914001.2, 4.4.68921.1, \(\Q(\sqrt{41}, \sqrt{-59})\), 8.0.2359907842908948041.2 x2, 8.4.677939627379761.3 x2, 8.0.57558727875828001.4

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
$59$59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$