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Properties

Label 16.0.10271172687...5625.8
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{10}$
Root discriminant $27.43$
Ramified primes $5, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6211, -1635, 26254, -24683, 43444, -30049, 28752, -14592, 10142, -4256, 2491, -896, 408, -100, 33, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 33*x^14 - 100*x^13 + 408*x^12 - 896*x^11 + 2491*x^10 - 4256*x^9 + 10142*x^8 - 14592*x^7 + 28752*x^6 - 30049*x^5 + 43444*x^4 - 24683*x^3 + 26254*x^2 - 1635*x + 6211)
 
gp: K = bnfinit(x^16 - 4*x^15 + 33*x^14 - 100*x^13 + 408*x^12 - 896*x^11 + 2491*x^10 - 4256*x^9 + 10142*x^8 - 14592*x^7 + 28752*x^6 - 30049*x^5 + 43444*x^4 - 24683*x^3 + 26254*x^2 - 1635*x + 6211, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 33 x^{14} - 100 x^{13} + 408 x^{12} - 896 x^{11} + 2491 x^{10} - 4256 x^{9} + 10142 x^{8} - 14592 x^{7} + 28752 x^{6} - 30049 x^{5} + 43444 x^{4} - 24683 x^{3} + 26254 x^{2} - 1635 x + 6211 \)

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:  \(102711726879931884765625=5^{12}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.43$
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:  $5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{71330222971495587143186174803439} a^{15} + \frac{16078391337243205803472185728312}{71330222971495587143186174803439} a^{14} + \frac{22975028017284727661215036061140}{71330222971495587143186174803439} a^{13} + \frac{24372000447809346799714831451708}{71330222971495587143186174803439} a^{12} - \frac{3729398107527778583265127942563}{71330222971495587143186174803439} a^{11} + \frac{24567846501539083758046935644983}{71330222971495587143186174803439} a^{10} - \frac{18107188632816452841928480973477}{71330222971495587143186174803439} a^{9} + \frac{31330057840261239210556054752413}{71330222971495587143186174803439} a^{8} - \frac{11552002783432123481267954456230}{71330222971495587143186174803439} a^{7} + \frac{10206227037863402279848902653019}{71330222971495587143186174803439} a^{6} - \frac{3087579958398767006947471673454}{71330222971495587143186174803439} a^{5} + \frac{25501090886766359265068839821916}{71330222971495587143186174803439} a^{4} + \frac{8070793013390314805673829387109}{71330222971495587143186174803439} a^{3} - \frac{17612800022040097523487565672470}{71330222971495587143186174803439} a^{2} + \frac{1140378114874698153915405775887}{71330222971495587143186174803439} a - \frac{18457393699154569367781207356042}{71330222971495587143186174803439}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 7760.72026823 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.105125.2, 4.4.725.1, 4.0.3625.1, 8.4.15243125.1, 8.4.320486703125.1, 8.0.11051265625.5

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$29$29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$