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Properties

Label 16.0.47496197427...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{8}\cdot 29^{6}\cdot 41^{8}$
Root discriminant $71.58$
Ramified primes $2, 5, 29, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T493)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12519376, 25811504, 17831808, 4003656, 129912, 234000, 13542, -10604, 22825, 4585, 1052, 757, 18, 24, 16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 16*x^14 + 24*x^13 + 18*x^12 + 757*x^11 + 1052*x^10 + 4585*x^9 + 22825*x^8 - 10604*x^7 + 13542*x^6 + 234000*x^5 + 129912*x^4 + 4003656*x^3 + 17831808*x^2 + 25811504*x + 12519376)
 
gp: K = bnfinit(x^16 - x^15 + 16*x^14 + 24*x^13 + 18*x^12 + 757*x^11 + 1052*x^10 + 4585*x^9 + 22825*x^8 - 10604*x^7 + 13542*x^6 + 234000*x^5 + 129912*x^4 + 4003656*x^3 + 17831808*x^2 + 25811504*x + 12519376, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 16 x^{14} + 24 x^{13} + 18 x^{12} + 757 x^{11} + 1052 x^{10} + 4585 x^{9} + 22825 x^{8} - 10604 x^{7} + 13542 x^{6} + 234000 x^{5} + 129912 x^{4} + 4003656 x^{3} + 17831808 x^{2} + 25811504 x + 12519376 \)

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:  \(474961974272243913084100000000=2^{8}\cdot 5^{8}\cdot 29^{6}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.58$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{3496} a^{14} - \frac{41}{3496} a^{13} - \frac{7}{92} a^{12} + \frac{159}{1748} a^{11} + \frac{185}{1748} a^{10} + \frac{545}{3496} a^{9} - \frac{153}{437} a^{8} - \frac{47}{152} a^{7} + \frac{1109}{3496} a^{6} - \frac{733}{1748} a^{5} + \frac{197}{437} a^{4} + \frac{160}{437} a^{3} + \frac{33}{437} a^{2} + \frac{211}{437} a + \frac{78}{437}$, $\frac{1}{22539869710928663087711679070412079234526232} a^{15} + \frac{470446204113287326499884736943571008857}{5634967427732165771927919767603019808631558} a^{14} + \frac{106469454749833126312791457628526449794977}{2049079064629878462519243551855643566775112} a^{13} - \frac{61683474149461013334365335865190134819935}{2817483713866082885963959883801509904315779} a^{12} + \frac{105481867052754486205348174142194442280503}{489997167628883980167645197182871287707092} a^{11} - \frac{4870572435073758334464301229116684662010549}{22539869710928663087711679070412079234526232} a^{10} + \frac{278555664307887341076299978612451315729123}{1186308932154140162511141003705898907080328} a^{9} - \frac{343069027381089761395708078146684861663217}{1186308932154140162511141003705898907080328} a^{8} - \frac{5224135423691678248303007319341897249643993}{11269934855464331543855839535206039617263116} a^{7} - \frac{4484489359183329965030857337953548688492007}{22539869710928663087711679070412079234526232} a^{6} + \frac{309477065401415793008417874411914094589241}{1024539532314939231259621775927821783387556} a^{5} - \frac{2694238680250436816969269354803435164248709}{11269934855464331543855839535206039617263116} a^{4} + \frac{17818469860256120983157555475392102755105}{5634967427732165771927919767603019808631558} a^{3} + \frac{2809310884037480534489867875707744809089145}{5634967427732165771927919767603019808631558} a^{2} + \frac{915626714259073868700949077341042731517863}{2817483713866082885963959883801509904315779} a - \frac{734620173514193493677522429476293331791303}{2817483713866082885963959883801509904315779}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 22333695.0148 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T493):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.29725.1, 4.0.1025.1, 4.4.725.1, 8.0.883575625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$