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Properties

Label 16.0.11447545997...1184.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{52}\cdot 3^{26}$
Root discriminant $56.71$
Ramified primes $2, 3$
Class number $72$ (GRH)
Class group $[2, 36]$ (GRH)
Galois group $\GL(2,Z/4)$ (as 16T193)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8395, -24824, 65712, -114520, 130226, -104280, 68860, -37448, 17271, -7040, 2284, -456, -22, 32, 12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 12*x^14 + 32*x^13 - 22*x^12 - 456*x^11 + 2284*x^10 - 7040*x^9 + 17271*x^8 - 37448*x^7 + 68860*x^6 - 104280*x^5 + 130226*x^4 - 114520*x^3 + 65712*x^2 - 24824*x + 8395)
 
gp: K = bnfinit(x^16 - 8*x^15 + 12*x^14 + 32*x^13 - 22*x^12 - 456*x^11 + 2284*x^10 - 7040*x^9 + 17271*x^8 - 37448*x^7 + 68860*x^6 - 104280*x^5 + 130226*x^4 - 114520*x^3 + 65712*x^2 - 24824*x + 8395, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 12 x^{14} + 32 x^{13} - 22 x^{12} - 456 x^{11} + 2284 x^{10} - 7040 x^{9} + 17271 x^{8} - 37448 x^{7} + 68860 x^{6} - 104280 x^{5} + 130226 x^{4} - 114520 x^{3} + 65712 x^{2} - 24824 x + 8395 \)

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:  \(11447545997288281555215581184=2^{52}\cdot 3^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.71$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{1}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{185094729} a^{14} + \frac{1934837}{185094729} a^{13} + \frac{3785203}{185094729} a^{12} - \frac{5554949}{185094729} a^{11} - \frac{23229295}{185094729} a^{10} - \frac{22233587}{185094729} a^{9} - \frac{1402535}{61698243} a^{8} + \frac{2243188}{20566081} a^{7} + \frac{1958697}{20566081} a^{6} + \frac{55315474}{185094729} a^{5} - \frac{56887552}{185094729} a^{4} - \frac{62766656}{185094729} a^{3} - \frac{49676591}{185094729} a^{2} + \frac{18539948}{185094729} a - \frac{50976347}{185094729}$, $\frac{1}{24440816302954553647521} a^{15} + \frac{20587946468714}{8146938767651517882507} a^{14} + \frac{5015327598395895235}{520017368147969226543} a^{13} + \frac{113626745796934122195}{2715646255883839294169} a^{12} + \frac{912079298116558699865}{24440816302954553647521} a^{11} - \frac{1156389541291464073288}{8146938767651517882507} a^{10} - \frac{1148052656068496332334}{24440816302954553647521} a^{9} + \frac{44500872115613120356}{2715646255883839294169} a^{8} + \frac{294870368446319037619}{8146938767651517882507} a^{7} - \frac{709936799472074107838}{24440816302954553647521} a^{6} + \frac{1657260338049263231387}{8146938767651517882507} a^{5} - \frac{6386356094354793539338}{24440816302954553647521} a^{4} + \frac{1676887576229310412882}{8146938767651517882507} a^{3} - \frac{5757555995143733041750}{24440816302954553647521} a^{2} + \frac{1004834619018082358033}{2715646255883839294169} a + \frac{5033202501652361138806}{24440816302954553647521}$

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

Class group and class number

$C_{2}\times C_{36}$, which has order $72$ (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:  \( 1249717.03243 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,Z/4)$ (as 16T193):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 14 conjugacy class representatives for $\GL(2,Z/4)$
Character table for $\GL(2,Z/4)$

Intermediate fields

\(\Q(\sqrt{6}) \), 4.4.497664.1, 4.0.27648.1, 8.8.2229025112064.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.12.23.50$x^{12} - 9 x^{11} + 6 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 9 x^{4} + 3 x^{3} - 9 x^{2} + 9 x + 3$$12$$1$$23$$(C_6\times C_2):C_2$$[5/2]_{4}^{2}$