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Properties

Label 16.0.22144295318...8144.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{12}\cdot 23^{6}$
Root discriminant $59.10$
Ramified primes $2, 3, 23$
Class number $128$ (GRH)
Class group $[2, 2, 4, 8]$ (GRH)
Galois group $C_4.D_4$ (as 16T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1377009, 1890576, 1807452, 850104, 39996, -191712, -11160, 65760, 20776, -8976, -4340, 696, 528, -24, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 - 24*x^13 + 528*x^12 + 696*x^11 - 4340*x^10 - 8976*x^9 + 20776*x^8 + 65760*x^7 - 11160*x^6 - 191712*x^5 + 39996*x^4 + 850104*x^3 + 1807452*x^2 + 1890576*x + 1377009)
 
gp: K = bnfinit(x^16 - 32*x^14 - 24*x^13 + 528*x^12 + 696*x^11 - 4340*x^10 - 8976*x^9 + 20776*x^8 + 65760*x^7 - 11160*x^6 - 191712*x^5 + 39996*x^4 + 850104*x^3 + 1807452*x^2 + 1890576*x + 1377009, 1)
 

Normalized defining polynomial

\( x^{16} - 32 x^{14} - 24 x^{13} + 528 x^{12} + 696 x^{11} - 4340 x^{10} - 8976 x^{9} + 20776 x^{8} + 65760 x^{7} - 11160 x^{6} - 191712 x^{5} + 39996 x^{4} + 850104 x^{3} + 1807452 x^{2} + 1890576 x + 1377009 \)

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:  \(22144295318673432094540038144=2^{48}\cdot 3^{12}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.10$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5}$, $\frac{1}{28977325491} a^{14} + \frac{630825893}{9659108497} a^{13} - \frac{808754470}{28977325491} a^{12} - \frac{2224212499}{9659108497} a^{11} + \frac{309255262}{28977325491} a^{10} + \frac{2753437559}{9659108497} a^{9} - \frac{227534638}{1259883717} a^{8} + \frac{2227491032}{9659108497} a^{7} + \frac{2012211509}{28977325491} a^{6} + \frac{391602657}{9659108497} a^{5} + \frac{429674876}{1259883717} a^{4} - \frac{3459708764}{9659108497} a^{3} + \frac{1449190320}{9659108497} a^{2} - \frac{1560940578}{9659108497} a + \frac{2126013471}{9659108497}$, $\frac{1}{3546282449623792190438960086114653} a^{15} - \frac{35445654927098724742595}{3546282449623792190438960086114653} a^{14} + \frac{170228537111504862929661206946622}{3546282449623792190438960086114653} a^{13} + \frac{53241585274898694909047116441772}{322389313602162926403541826010423} a^{12} - \frac{53254462517771133543098961437273}{107463104534054308801180608670141} a^{11} - \frac{204753303963383621555884619450935}{1182094149874597396812986695371551} a^{10} + \frac{502773713211026098137071710134415}{3546282449623792190438960086114653} a^{9} - \frac{771570633547256819313095082844532}{3546282449623792190438960086114653} a^{8} + \frac{1534246715619917735580026110780942}{3546282449623792190438960086114653} a^{7} + \frac{35925785723092955190279141477794}{322389313602162926403541826010423} a^{6} + \frac{475164052328543320810482539456715}{1182094149874597396812986695371551} a^{5} + \frac{115294702891750668869691636325840}{1182094149874597396812986695371551} a^{4} + \frac{144895358557628862227045511047629}{1182094149874597396812986695371551} a^{3} - \frac{447679619259919041834523799681624}{1182094149874597396812986695371551} a^{2} - \frac{5185700149889489265064126841749}{1182094149874597396812986695371551} a - \frac{377046843013254862436815777571760}{1182094149874597396812986695371551}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{8}$, which has order $128$ (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:  \( 475824.014517 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4$ (as 16T30):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4.D_4$
Character table for $C_4.D_4$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.0.105984.1, 4.0.105984.2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.404373897216.34, 8.0.179721732096.5, 8.8.1617495588864.1

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 16 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$