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Properties

Label 16.0.14013832946...1216.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{56}\cdot 3^{4}\cdot 7^{4}$
Root discriminant $24.22$
Ramified primes $2, 3, 7$
Class number $2$
Class group $[2]$
Galois group $C_2^4.C_2^4$ (as 16T459)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34, -144, 496, -1120, 2100, -3056, 3768, -3808, 3342, -2520, 1672, -944, 440, -160, 44, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 44*x^14 - 160*x^13 + 440*x^12 - 944*x^11 + 1672*x^10 - 2520*x^9 + 3342*x^8 - 3808*x^7 + 3768*x^6 - 3056*x^5 + 2100*x^4 - 1120*x^3 + 496*x^2 - 144*x + 34)
 
gp: K = bnfinit(x^16 - 8*x^15 + 44*x^14 - 160*x^13 + 440*x^12 - 944*x^11 + 1672*x^10 - 2520*x^9 + 3342*x^8 - 3808*x^7 + 3768*x^6 - 3056*x^5 + 2100*x^4 - 1120*x^3 + 496*x^2 - 144*x + 34, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 44 x^{14} - 160 x^{13} + 440 x^{12} - 944 x^{11} + 1672 x^{10} - 2520 x^{9} + 3342 x^{8} - 3808 x^{7} + 3768 x^{6} - 3056 x^{5} + 2100 x^{4} - 1120 x^{3} + 496 x^{2} - 144 x + 34 \)

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:  \(14013832946090262921216=2^{56}\cdot 3^{4}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.22$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{204924499740093} a^{15} - \frac{4449878187791}{68308166580031} a^{14} + \frac{27209164096714}{68308166580031} a^{13} - \frac{29854524090340}{68308166580031} a^{12} - \frac{145236580890}{68308166580031} a^{11} - \frac{23933740671583}{204924499740093} a^{10} + \frac{31024043188080}{68308166580031} a^{9} + \frac{21864113720390}{68308166580031} a^{8} + \frac{4529231209788}{9758309511433} a^{7} + \frac{1188730439078}{4182132647757} a^{6} + \frac{81407258651161}{204924499740093} a^{5} - \frac{58783438252876}{204924499740093} a^{4} - \frac{96355223156624}{204924499740093} a^{3} + \frac{3953767071563}{68308166580031} a^{2} - \frac{6424514145479}{29274928534299} a - \frac{87536696754532}{204924499740093}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{18317116649378}{204924499740093} a^{15} - \frac{177204082435229}{204924499740093} a^{14} + \frac{1027393810569526}{204924499740093} a^{13} - \frac{4084957416270389}{204924499740093} a^{12} + \frac{11894116659636856}{204924499740093} a^{11} - \frac{26871668791010884}{204924499740093} a^{10} + \frac{16295138157757366}{68308166580031} a^{9} - \frac{24916430381302029}{68308166580031} a^{8} + \frac{4720055143811040}{9758309511433} a^{7} - \frac{2333248856395124}{4182132647757} a^{6} + \frac{111142277185738108}{204924499740093} a^{5} - \frac{89193202437986417}{204924499740093} a^{4} + \frac{54825599508108676}{204924499740093} a^{3} - \frac{8917461395939130}{68308166580031} a^{2} + \frac{1229531722538816}{29274928534299} a - \frac{2088014619722833}{204924499740093} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 132358.668627 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^4$ (as 16T459):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^4.C_2^4$
Character table for $C_2^4.C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.7168.1, 4.4.7168.1, \(\Q(\zeta_{8})\), 8.0.205520896.4

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$