↑

Properties

Label 16.4.24109722907...601.42
Degree $16$
Signature $[4, 6]$
Discriminant $13^{10}\cdot 53^{10}$
Root discriminant $59.41$
Ramified primes $13, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T875

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![608357, -1315048, 712532, 197845, -201687, -21992, 27073, -6056, 3409, 1073, -744, -211, 80, 1, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 3*x^14 + x^13 + 80*x^12 - 211*x^11 - 744*x^10 + 1073*x^9 + 3409*x^8 - 6056*x^7 + 27073*x^6 - 21992*x^5 - 201687*x^4 + 197845*x^3 + 712532*x^2 - 1315048*x + 608357)
 
gp: K = bnfinit(x^16 - x^15 - 3*x^14 + x^13 + 80*x^12 - 211*x^11 - 744*x^10 + 1073*x^9 + 3409*x^8 - 6056*x^7 + 27073*x^6 - 21992*x^5 - 201687*x^4 + 197845*x^3 + 712532*x^2 - 1315048*x + 608357, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 3 x^{14} + x^{13} + 80 x^{12} - 211 x^{11} - 744 x^{10} + 1073 x^{9} + 3409 x^{8} - 6056 x^{7} + 27073 x^{6} - 21992 x^{5} - 201687 x^{4} + 197845 x^{3} + 712532 x^{2} - 1315048 x + 608357 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24109722907876309716269637601=13^{10}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.41$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{318} a^{14} - \frac{11}{106} a^{13} + \frac{1}{159} a^{12} - \frac{7}{53} a^{11} - \frac{7}{53} a^{10} + \frac{119}{318} a^{9} - \frac{1}{318} a^{8} + \frac{28}{159} a^{7} + \frac{62}{159} a^{6} + \frac{21}{53} a^{5} - \frac{39}{106} a^{4} - \frac{101}{318} a^{3} - \frac{61}{159} a^{2} + \frac{38}{159} a + \frac{37}{159}$, $\frac{1}{330801524728533909229941873151379802} a^{15} - \frac{250782954994196665801980928133341}{165400762364266954614970936575689901} a^{14} + \frac{10521448929154367834074826982882961}{165400762364266954614970936575689901} a^{13} + \frac{8929202053327387970920557734141023}{165400762364266954614970936575689901} a^{12} - \frac{19318034888422552955988036943121653}{110267174909511303076647291050459934} a^{11} + \frac{2407316755294908806581974287610067}{165400762364266954614970936575689901} a^{10} - \frac{30799496129719793772629291453883229}{165400762364266954614970936575689901} a^{9} - \frac{69896729290371924145149379786645321}{165400762364266954614970936575689901} a^{8} - \frac{26992245346028308145685987954919115}{55133587454755651538323645525229967} a^{7} + \frac{55199168035746556270207605767142367}{330801524728533909229941873151379802} a^{6} - \frac{634991298087054531969850034487577}{18377862484918550512774548508409989} a^{5} + \frac{65217768919271800112140752819263222}{165400762364266954614970936575689901} a^{4} + \frac{34079100289374559676210369639333092}{165400762364266954614970936575689901} a^{3} + \frac{34028822735328584848209279830011189}{165400762364266954614970936575689901} a^{2} + \frac{24188077365343569787300750317389789}{110267174909511303076647291050459934} a - \frac{128841093702686462188697060700334771}{330801524728533909229941873151379802}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 116389821.925 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.225360027841.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
53Data not computed