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Properties

Label 16.16.7213895789...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{50}\cdot 3^{8}\cdot 5^{10}$
Root discriminant $41.32$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^2.D_4$ (as 16T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 136, 1172, 1464, -6596, -3184, 12708, -344, -9507, 2824, 2568, -1168, -230, 168, -4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 230*x^12 - 1168*x^11 + 2568*x^10 + 2824*x^9 - 9507*x^8 - 344*x^7 + 12708*x^6 - 3184*x^5 - 6596*x^4 + 1464*x^3 + 1172*x^2 + 136*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 - 4*x^14 + 168*x^13 - 230*x^12 - 1168*x^11 + 2568*x^10 + 2824*x^9 - 9507*x^8 - 344*x^7 + 12708*x^6 - 3184*x^5 - 6596*x^4 + 1464*x^3 + 1172*x^2 + 136*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 4 x^{14} + 168 x^{13} - 230 x^{12} - 1168 x^{11} + 2568 x^{10} + 2824 x^{9} - 9507 x^{8} - 344 x^{7} + 12708 x^{6} - 3184 x^{5} - 6596 x^{4} + 1464 x^{3} + 1172 x^{2} + 136 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(72138957898383360000000000=2^{50}\cdot 3^{8}\cdot 5^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.32$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5}$, $\frac{1}{96195} a^{14} - \frac{7}{96195} a^{13} + \frac{2437}{32065} a^{12} - \frac{5297}{96195} a^{11} + \frac{247}{8745} a^{10} + \frac{83}{2915} a^{9} + \frac{1211}{19239} a^{8} + \frac{36038}{96195} a^{7} - \frac{3893}{96195} a^{6} + \frac{13864}{32065} a^{5} - \frac{124}{795} a^{4} + \frac{3544}{96195} a^{3} - \frac{510}{6413} a^{2} - \frac{10429}{96195} a - \frac{37067}{96195}$, $\frac{1}{347744925} a^{15} + \frac{8}{1545533} a^{14} + \frac{17425196}{347744925} a^{13} - \frac{30582284}{347744925} a^{12} - \frac{2955053}{38638325} a^{11} + \frac{3727}{54225} a^{10} - \frac{4529399}{347744925} a^{9} + \frac{7705744}{115914975} a^{8} + \frac{38950148}{115914975} a^{7} + \frac{173526478}{347744925} a^{6} - \frac{14165698}{347744925} a^{5} + \frac{51784664}{115914975} a^{4} + \frac{127447}{1312245} a^{3} - \frac{64201681}{347744925} a^{2} + \frac{4847378}{10537725} a + \frac{103150408}{347744925}$

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:  $15$
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:  \( 84840645.5995 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2.D_4$ (as 16T92):

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 $C_2\times C_2^2.D_4$
Character table for $C_2\times C_2^2.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{5})\), 8.8.8493465600000.2, 8.8.3317760000.1, 8.8.8493465600000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$