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Properties

Label 16.0.88208360493...5216.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 19^{10}\cdot 37^{8}$
Root discriminant $64.43$
Ramified primes $2, 19, 37$
Class number $340$ (GRH)
Class group $[2, 170]$ (GRH)
Galois group $C_2^3.Q_8.C_6$ (as 16T732)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![160747, -515639, 396174, -105096, 303508, -136431, 80892, -49147, 31473, -13083, 6112, -2227, 702, -176, 38, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 38*x^14 - 176*x^13 + 702*x^12 - 2227*x^11 + 6112*x^10 - 13083*x^9 + 31473*x^8 - 49147*x^7 + 80892*x^6 - 136431*x^5 + 303508*x^4 - 105096*x^3 + 396174*x^2 - 515639*x + 160747)
 
gp: K = bnfinit(x^16 - 7*x^15 + 38*x^14 - 176*x^13 + 702*x^12 - 2227*x^11 + 6112*x^10 - 13083*x^9 + 31473*x^8 - 49147*x^7 + 80892*x^6 - 136431*x^5 + 303508*x^4 - 105096*x^3 + 396174*x^2 - 515639*x + 160747, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 38 x^{14} - 176 x^{13} + 702 x^{12} - 2227 x^{11} + 6112 x^{10} - 13083 x^{9} + 31473 x^{8} - 49147 x^{7} + 80892 x^{6} - 136431 x^{5} + 303508 x^{4} - 105096 x^{3} + 396174 x^{2} - 515639 x + 160747 \)

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:  \(88208360493688117191834505216=2^{12}\cdot 19^{10}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.43$
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:  $2, 19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a$, $\frac{1}{36068687001346242612365768115753859778476} a^{15} - \frac{2085191185367403461983860928415829686343}{18034343500673121306182884057876929889238} a^{14} - \frac{1493123437605938625550117442502397713073}{36068687001346242612365768115753859778476} a^{13} + \frac{616470175495707921019277117778109067186}{9017171750336560653091442028938464944619} a^{12} + \frac{3323607676296434150185641179762630110275}{36068687001346242612365768115753859778476} a^{11} + \frac{3274808476248421900769565883657966828933}{36068687001346242612365768115753859778476} a^{10} + \frac{4609277029691068040654044648747613030217}{36068687001346242612365768115753859778476} a^{9} + \frac{5483014540068282609935448621840556012161}{36068687001346242612365768115753859778476} a^{8} + \frac{2392751284790289491078639156118424640659}{36068687001346242612365768115753859778476} a^{7} + \frac{4098578961433006956602961121820741740065}{36068687001346242612365768115753859778476} a^{6} + \frac{9705017776613926262644987849132254554333}{36068687001346242612365768115753859778476} a^{5} - \frac{2832331847650040381498451197649642648277}{36068687001346242612365768115753859778476} a^{4} + \frac{7800993776995670773768045201257674596035}{18034343500673121306182884057876929889238} a^{3} + \frac{4869605224276404235203123043238336531045}{36068687001346242612365768115753859778476} a^{2} + \frac{3954896200771658589346249606691840668465}{18034343500673121306182884057876929889238} a + \frac{686352816779603685633632590429763971249}{1568203782667227939668076874597993903412}$

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

Class group and class number

$C_{2}\times C_{170}$, which has order $340$ (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:  \( 168824.791868 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.Q_8.C_6$ (as 16T732):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 30 conjugacy class representatives for $C_2^3.Q_8.C_6$
Character table for $C_2^3.Q_8.C_6$ is not computed

Intermediate fields

4.4.494209.1, 8.8.244242535681.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: data not computed
Degree 32 sibling: 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.6.4.2$x^{6} - 19 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.2$x^{6} - 19 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.12.8.1$x^{12} - 111 x^{9} + 4107 x^{6} - 50653 x^{3} + 14993288$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$