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Properties

Label 16.0.73146118344...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $55.15$
Ramified primes $2, 5, 17$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $(C_2\times D_4):C_4$ (as 16T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![456641, -208444, -429022, 360200, 163690, -203252, -11840, 58136, -3489, -8140, 80, 784, 218, -64, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 18*x^14 - 64*x^13 + 218*x^12 + 784*x^11 + 80*x^10 - 8140*x^9 - 3489*x^8 + 58136*x^7 - 11840*x^6 - 203252*x^5 + 163690*x^4 + 360200*x^3 - 429022*x^2 - 208444*x + 456641)
 
gp: K = bnfinit(x^16 - 18*x^14 - 64*x^13 + 218*x^12 + 784*x^11 + 80*x^10 - 8140*x^9 - 3489*x^8 + 58136*x^7 - 11840*x^6 - 203252*x^5 + 163690*x^4 + 360200*x^3 - 429022*x^2 - 208444*x + 456641, 1)
 

Normalized defining polynomial

\( x^{16} - 18 x^{14} - 64 x^{13} + 218 x^{12} + 784 x^{11} + 80 x^{10} - 8140 x^{9} - 3489 x^{8} + 58136 x^{7} - 11840 x^{6} - 203252 x^{5} + 163690 x^{4} + 360200 x^{3} - 429022 x^{2} - 208444 x + 456641 \)

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:  \(7314611834454016000000000000=2^{32}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.15$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{10} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{5969657553481474528026143583280494294} a^{15} + \frac{350532255909309783414016775261054629}{5969657553481474528026143583280494294} a^{14} - \frac{140490061349596876994679601962388943}{2984828776740737264013071791640247147} a^{13} - \frac{249535319820318532660259689568952587}{5969657553481474528026143583280494294} a^{12} + \frac{997278968719642799789652549049871879}{5969657553481474528026143583280494294} a^{11} + \frac{1005348952452349526371613651509130461}{5969657553481474528026143583280494294} a^{10} - \frac{743211337160601625529383901698559845}{5969657553481474528026143583280494294} a^{9} + \frac{435561666888864828965341917067290817}{1989885851160491509342047861093498098} a^{8} + \frac{2018662935048343012837853456717355479}{5969657553481474528026143583280494294} a^{7} + \frac{105494267925477105892174279575477907}{5969657553481474528026143583280494294} a^{6} + \frac{472346707705352684846696519425875405}{1989885851160491509342047861093498098} a^{5} - \frac{740402323538246292650469881171836425}{1989885851160491509342047861093498098} a^{4} + \frac{125200984047779827133820367841997143}{994942925580245754671023930546749049} a^{3} + \frac{630117748120076185607334061418807408}{2984828776740737264013071791640247147} a^{2} - \frac{1531055422467026234351097301395057425}{5969657553481474528026143583280494294} a + \frac{194935371225303653131195906463180648}{2984828776740737264013071791640247147}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 2872553.41913 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4):C_4$ (as 16T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times D_4):C_4$
Character table for $(C_2\times D_4):C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), 4.0.27200.2, 4.0.1088.2, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.739840000.6

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
5Data not computed
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$