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Properties

Label 16.4.10895198697...3125.2
Degree $16$
Signature $[4, 6]$
Discriminant $5^{10}\cdot 101^{11}$
Root discriminant $65.29$
Ramified primes $5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-165475, 532775, -641275, 362275, -96760, -45115, 113935, -68780, -6635, 20100, -3805, -2060, 677, 85, -43, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 43*x^14 + 85*x^13 + 677*x^12 - 2060*x^11 - 3805*x^10 + 20100*x^9 - 6635*x^8 - 68780*x^7 + 113935*x^6 - 45115*x^5 - 96760*x^4 + 362275*x^3 - 641275*x^2 + 532775*x - 165475)
 
gp: K = bnfinit(x^16 - x^15 - 43*x^14 + 85*x^13 + 677*x^12 - 2060*x^11 - 3805*x^10 + 20100*x^9 - 6635*x^8 - 68780*x^7 + 113935*x^6 - 45115*x^5 - 96760*x^4 + 362275*x^3 - 641275*x^2 + 532775*x - 165475, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 43 x^{14} + 85 x^{13} + 677 x^{12} - 2060 x^{11} - 3805 x^{10} + 20100 x^{9} - 6635 x^{8} - 68780 x^{7} + 113935 x^{6} - 45115 x^{5} - 96760 x^{4} + 362275 x^{3} - 641275 x^{2} + 532775 x - 165475 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(108951986979034819834970703125=5^{10}\cdot 101^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{8}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8}$, $\frac{1}{4829698528029381940426036435} a^{15} - \frac{7295788895805461876361049}{965939705605876388085207287} a^{14} - \frac{43605934064118568788027696}{965939705605876388085207287} a^{13} + \frac{246088634194830755014451472}{4829698528029381940426036435} a^{12} + \frac{6494110523259475228084697}{209986892523016606105479845} a^{11} - \frac{21347784761800709438675115}{965939705605876388085207287} a^{10} - \frac{1783193486709659112881744287}{4829698528029381940426036435} a^{9} - \frac{1439016893633905201736206907}{4829698528029381940426036435} a^{8} - \frac{314918019563823241255153591}{965939705605876388085207287} a^{7} - \frac{270835423320605025798934234}{965939705605876388085207287} a^{6} + \frac{58374052076397463842789562}{965939705605876388085207287} a^{5} + \frac{470403459222080378464831298}{965939705605876388085207287} a^{4} + \frac{361918206732972069887538391}{965939705605876388085207287} a^{3} - \frac{251380987522379025651276758}{965939705605876388085207287} a^{2} + \frac{16196375058396569557374801}{41997378504603321221095969} a + \frac{127490135210825902821783431}{965939705605876388085207287}$

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:  $9$
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:  \( 257478301.155 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.51005.1, 8.4.6568812813125.3

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 $16$ $16$ R $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
101Data not computed