↑

Properties

Label 16.8.83120476897...3681.2
Degree $16$
Signature $[8, 4]$
Discriminant $23^{12}\cdot 41^{14}$
Root discriminant $270.70$
Ramified primes $23, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5374176512, 3767796992, -6683215296, 269660476, -24483833, -42000364, 59898657, -4611138, -348651, 272878, -99203, 8650, -2072, -286, 90, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 90*x^14 - 286*x^13 - 2072*x^12 + 8650*x^11 - 99203*x^10 + 272878*x^9 - 348651*x^8 - 4611138*x^7 + 59898657*x^6 - 42000364*x^5 - 24483833*x^4 + 269660476*x^3 - 6683215296*x^2 + 3767796992*x + 5374176512)
 
gp: K = bnfinit(x^16 - 2*x^15 + 90*x^14 - 286*x^13 - 2072*x^12 + 8650*x^11 - 99203*x^10 + 272878*x^9 - 348651*x^8 - 4611138*x^7 + 59898657*x^6 - 42000364*x^5 - 24483833*x^4 + 269660476*x^3 - 6683215296*x^2 + 3767796992*x + 5374176512, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 90 x^{14} - 286 x^{13} - 2072 x^{12} + 8650 x^{11} - 99203 x^{10} + 272878 x^{9} - 348651 x^{8} - 4611138 x^{7} + 59898657 x^{6} - 42000364 x^{5} - 24483833 x^{4} + 269660476 x^{3} - 6683215296 x^{2} + 3767796992 x + 5374176512 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(831204768973346090928597984067970713681=23^{12}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $270.70$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{944} a^{14} - \frac{55}{472} a^{13} + \frac{31}{472} a^{12} - \frac{29}{472} a^{11} - \frac{9}{59} a^{10} + \frac{11}{472} a^{9} + \frac{113}{944} a^{8} + \frac{57}{472} a^{7} + \frac{421}{944} a^{6} + \frac{53}{472} a^{5} - \frac{307}{944} a^{4} - \frac{39}{236} a^{3} + \frac{195}{944} a^{2} - \frac{21}{118} a + \frac{26}{59}$, $\frac{1}{509032613714421296422140438830471539673459454460611293085227746497088} a^{15} + \frac{1986345009741155877474973488053308242869952001462327642178426513}{4313835709444248274763902023987046946385249614072977060044302936416} a^{14} - \frac{1726796668318128278729972450737466772415983717108253348744661154011}{254516306857210648211070219415235769836729727230305646542613873248544} a^{13} + \frac{12206940095194119067989210007232861034453747330915354531857024577633}{254516306857210648211070219415235769836729727230305646542613873248544} a^{12} + \frac{4932918720929196167644354486119393320678250749136101220119993723367}{63629076714302662052767554853808942459182431807576411635653468312136} a^{11} + \frac{59259457373230287106542155431584599015055441814748917641102204947741}{254516306857210648211070219415235769836729727230305646542613873248544} a^{10} + \frac{80136783186764315877492209985533148597191644703736770618507288675677}{509032613714421296422140438830471539673459454460611293085227746497088} a^{9} - \frac{21705359483683919000004763723737292703531138407569372344861012098605}{254516306857210648211070219415235769836729727230305646542613873248544} a^{8} + \frac{23151200003366166011351206099555706867565601941328021331655729714357}{509032613714421296422140438830471539673459454460611293085227746497088} a^{7} - \frac{43786270250021244087833603639224831065914083884239041136047545347541}{254516306857210648211070219415235769836729727230305646542613873248544} a^{6} - \frac{192082940320750127996694742917226339319944588237988666983292592480399}{509032613714421296422140438830471539673459454460611293085227746497088} a^{5} + \frac{14913436100879644550575342878069576648521965878069245832040641893759}{127258153428605324105535109707617884918364863615152823271306936624272} a^{4} - \frac{77704979492503303144810563470144834752765652936848054087250468655769}{509032613714421296422140438830471539673459454460611293085227746497088} a^{3} - \frac{930616272567331965515986314802569650054044999211331402251677390311}{127258153428605324105535109707617884918364863615152823271306936624272} a^{2} + \frac{2188714497133286554815060570875156641566208285092631164393969097751}{31814538357151331026383777426904471229591215903788205817826734156068} a - \frac{2142080464749929686446109829063840662618536825077334966063234365079}{7953634589287832756595944356726117807397803975947051454456683539017}$

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:  $11$
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:  \( 9032154107020 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$