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Properties

Label 16.0.19294036897...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{14}\cdot 29^{6}$
Root discriminant $32.95$
Ramified primes $3, 5, 29$
Class number $32$ (GRH)
Class group $[2, 4, 4]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48151, -9291, 15096, 1706, -187, 2529, -3546, 513, -912, -181, 396, -117, 183, -32, 24, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 24*x^14 - 32*x^13 + 183*x^12 - 117*x^11 + 396*x^10 - 181*x^9 - 912*x^8 + 513*x^7 - 3546*x^6 + 2529*x^5 - 187*x^4 + 1706*x^3 + 15096*x^2 - 9291*x + 48151)
 
gp: K = bnfinit(x^16 - 2*x^15 + 24*x^14 - 32*x^13 + 183*x^12 - 117*x^11 + 396*x^10 - 181*x^9 - 912*x^8 + 513*x^7 - 3546*x^6 + 2529*x^5 - 187*x^4 + 1706*x^3 + 15096*x^2 - 9291*x + 48151, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 24 x^{14} - 32 x^{13} + 183 x^{12} - 117 x^{11} + 396 x^{10} - 181 x^{9} - 912 x^{8} + 513 x^{7} - 3546 x^{6} + 2529 x^{5} - 187 x^{4} + 1706 x^{3} + 15096 x^{2} - 9291 x + 48151 \)

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:  \(1929403689792242431640625=3^{12}\cdot 5^{14}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.95$
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:  $3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{3}{11} a^{13} + \frac{5}{11} a^{11} + \frac{2}{11} a^{10} - \frac{1}{11} a^{9} + \frac{2}{11} a^{8} - \frac{2}{11} a^{7} + \frac{4}{11} a^{6} + \frac{2}{11} a^{5} - \frac{4}{11} a^{4} + \frac{4}{11} a^{3} + \frac{5}{11} a^{2} - \frac{2}{11} a + \frac{3}{11}$, $\frac{1}{163716934823983705957033413481113431} a^{15} - \frac{4901717723355310830909909706754765}{163716934823983705957033413481113431} a^{14} + \frac{16816806431252395498875536321456511}{163716934823983705957033413481113431} a^{13} + \frac{63708225384737385434317684983517493}{163716934823983705957033413481113431} a^{12} + \frac{78885178012535503821465031784676846}{163716934823983705957033413481113431} a^{11} - \frac{42292832285622196330042619384996}{513219231423146413658411954486249} a^{10} - \frac{50742254055393067388038387768286103}{163716934823983705957033413481113431} a^{9} - \frac{36584063614633894992035651479613}{5645411545654610550242531499348739} a^{8} + \frac{38361337299283193872655679801326735}{163716934823983705957033413481113431} a^{7} + \frac{15858682758840706188602098101701757}{163716934823983705957033413481113431} a^{6} - \frac{25452829778756324379294202676553905}{163716934823983705957033413481113431} a^{5} + \frac{76541406062337325766435325825793209}{163716934823983705957033413481113431} a^{4} - \frac{80135555975701164710018689730745527}{163716934823983705957033413481113431} a^{3} + \frac{73028636432580674062554750538212434}{163716934823983705957033413481113431} a^{2} + \frac{35355266840193783257183449845788150}{163716934823983705957033413481113431} a - \frac{7554416834559473221892281572469865}{163716934823983705957033413481113431}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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:  \( 6317.46461709 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 4.4.32625.1, 4.4.725.1, 8.0.47897578125.1 x2, 8.8.1064390625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
5Data not computed
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$