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Properties

Label 16.8.24664724269...0625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 29^{6}\cdot 101^{6}$
Root discriminant $44.62$
Ramified primes $5, 29, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2.C_2^2:D_4$ (as 16T225)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-409, 3873, -13023, 22388, -28295, 24362, -13772, 4693, 2268, -2499, 1258, -284, -201, 114, -3, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 3*x^14 + 114*x^13 - 201*x^12 - 284*x^11 + 1258*x^10 - 2499*x^9 + 2268*x^8 + 4693*x^7 - 13772*x^6 + 24362*x^5 - 28295*x^4 + 22388*x^3 - 13023*x^2 + 3873*x - 409)
 
gp: K = bnfinit(x^16 - 7*x^15 - 3*x^14 + 114*x^13 - 201*x^12 - 284*x^11 + 1258*x^10 - 2499*x^9 + 2268*x^8 + 4693*x^7 - 13772*x^6 + 24362*x^5 - 28295*x^4 + 22388*x^3 - 13023*x^2 + 3873*x - 409, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 3 x^{14} + 114 x^{13} - 201 x^{12} - 284 x^{11} + 1258 x^{10} - 2499 x^{9} + 2268 x^{8} + 4693 x^{7} - 13772 x^{6} + 24362 x^{5} - 28295 x^{4} + 22388 x^{3} - 13023 x^{2} + 3873 x - 409 \)

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:  \(246647242690979283562890625=5^{8}\cdot 29^{6}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.62$
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:  $5, 29, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{92} a^{14} + \frac{7}{92} a^{13} + \frac{5}{46} a^{12} - \frac{19}{92} a^{11} + \frac{17}{92} a^{10} - \frac{41}{92} a^{9} + \frac{21}{92} a^{8} - \frac{2}{23} a^{7} - \frac{43}{92} a^{6} + \frac{33}{92} a^{5} - \frac{41}{92} a^{4} - \frac{39}{92} a^{3} + \frac{9}{23} a^{2} + \frac{33}{92} a - \frac{27}{92}$, $\frac{1}{197737582805028802572935272} a^{15} - \frac{58992138551682974577216}{24717197850628600321616909} a^{14} - \frac{37054012761425190131233635}{197737582805028802572935272} a^{13} + \frac{29593159341703937418437}{781571473537663251276424} a^{12} - \frac{17277872257807168083223891}{98868791402514401286467636} a^{11} - \frac{21009776966320251370625333}{98868791402514401286467636} a^{10} + \frac{28771914424792630785000}{165887233896836243769241} a^{9} - \frac{14289432894769714596100779}{197737582805028802572935272} a^{8} - \frac{87274314815087164398929357}{197737582805028802572935272} a^{7} + \frac{34032618918149783455797011}{98868791402514401286467636} a^{6} + \frac{47004020928509823804361245}{98868791402514401286467636} a^{5} - \frac{17449522816144649749791277}{49434395701257200643233818} a^{4} - \frac{77917300258750163720917979}{197737582805028802572935272} a^{3} - \frac{78654734576167670905210397}{197737582805028802572935272} a^{2} + \frac{56400343786379466569223}{8988071945683127389678876} a + \frac{10190064533045512181272867}{197737582805028802572935272}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 8778050.93629 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^2:D_4$ (as 16T225):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^2.C_2^2:D_4$
Character table for $C_2^2.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 4.4.73225.1, 4.4.725.1, 8.8.5361900625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
101Data not computed