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Properties

Label 16.0.79688620999...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 29^{4}\cdot 181^{4}$
Root discriminant $98.59$
Ramified primes $2, 3, 5, 29, 181$
Class number $446080$ (GRH)
Class group $[2, 2, 2, 2, 27880]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T203)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18671230096, 10096911248, 35038013272, -10291651056, 10813266968, -2362029700, 1308764896, -201140128, 77623693, -8510506, 2417269, -185618, 39783, -1972, 323, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 323*x^14 - 1972*x^13 + 39783*x^12 - 185618*x^11 + 2417269*x^10 - 8510506*x^9 + 77623693*x^8 - 201140128*x^7 + 1308764896*x^6 - 2362029700*x^5 + 10813266968*x^4 - 10291651056*x^3 + 35038013272*x^2 + 10096911248*x + 18671230096)
 
gp: K = bnfinit(x^16 - 8*x^15 + 323*x^14 - 1972*x^13 + 39783*x^12 - 185618*x^11 + 2417269*x^10 - 8510506*x^9 + 77623693*x^8 - 201140128*x^7 + 1308764896*x^6 - 2362029700*x^5 + 10813266968*x^4 - 10291651056*x^3 + 35038013272*x^2 + 10096911248*x + 18671230096, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 323 x^{14} - 1972 x^{13} + 39783 x^{12} - 185618 x^{11} + 2417269 x^{10} - 8510506 x^{9} + 77623693 x^{8} - 201140128 x^{7} + 1308764896 x^{6} - 2362029700 x^{5} + 10813266968 x^{4} - 10291651056 x^{3} + 35038013272 x^{2} + 10096911248 x + 18671230096 \)

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:  \(79688620999701608976000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 29^{4}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.59$
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:  $2, 3, 5, 29, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} + \frac{1}{4} a^{9} - \frac{3}{8} a^{8} + \frac{1}{4} a^{7} - \frac{3}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{20078743736953765751769394435989856105619144728806529678226793244081674248} a^{15} + \frac{143536511391785068684320196127556934060189082119829707023006889173113205}{20078743736953765751769394435989856105619144728806529678226793244081674248} a^{14} + \frac{36469185243202775039273946127844117897608220452520268240998334114552177}{20078743736953765751769394435989856105619144728806529678226793244081674248} a^{13} + \frac{2158165650470015886532103139375534338666552499986561746651216637360600611}{20078743736953765751769394435989856105619144728806529678226793244081674248} a^{12} + \frac{4074294105897436878523542226252175081963088979236304068669849878527536617}{20078743736953765751769394435989856105619144728806529678226793244081674248} a^{11} - \frac{422287847699579514720688960064484528215313063118986854991989846282010315}{20078743736953765751769394435989856105619144728806529678226793244081674248} a^{10} + \frac{301548627916473367759772998343182773410811884002136135862921026493632395}{1056775986155461355356283917683676637137849722568764719906673328635877592} a^{9} + \frac{4807911632665900349010757157558578093878469492920810392531783810420476855}{20078743736953765751769394435989856105619144728806529678226793244081674248} a^{8} - \frac{1093038879225817148962370085037492709621056661524182729389278172149215651}{20078743736953765751769394435989856105619144728806529678226793244081674248} a^{7} + \frac{3511837226972635585167469033772489447702340916840139383313522593424960057}{20078743736953765751769394435989856105619144728806529678226793244081674248} a^{6} + \frac{4000042114462689817936931215523721217822696358126850744769684835220006617}{10039371868476882875884697217994928052809572364403264839113396622040837124} a^{5} + \frac{59671970596605977235982392098883928022154513570123733568925412248344343}{2509842967119220718971174304498732013202393091100816209778349155510209281} a^{4} - \frac{647858891644210580332485008433821514812235778243674705516140686459709326}{2509842967119220718971174304498732013202393091100816209778349155510209281} a^{3} - \frac{124716932602292592481431659640852548873502308507021355508703666867734547}{264193996538865338839070979420919159284462430642191179976668332158969398} a^{2} + \frac{1238319750441819482693875799675796394727630690930601670924312215702133101}{2509842967119220718971174304498732013202393091100816209778349155510209281} a - \frac{7725508237771943513869696639947822567212691667546583824189940445204089}{61215682124859041926126202548749561297619343685385761214106076963663641}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{27880}$, which has order $446080$ (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:  \( 3121.7160225 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T203):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 41 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{60})^+\)

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$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}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
181Data not computed