↑

Properties

Label 16.0.25694533210...3389.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{10}\cdot 61^{11}$
Root discriminant $33.54$
Ramified primes $3, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![85039, -49215, 96634, -79970, 136834, -91886, 91545, -47785, 31044, -12483, 5897, -1757, 641, -130, 38, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 38*x^14 - 130*x^13 + 641*x^12 - 1757*x^11 + 5897*x^10 - 12483*x^9 + 31044*x^8 - 47785*x^7 + 91545*x^6 - 91886*x^5 + 136834*x^4 - 79970*x^3 + 96634*x^2 - 49215*x + 85039)
 
gp: K = bnfinit(x^16 - 4*x^15 + 38*x^14 - 130*x^13 + 641*x^12 - 1757*x^11 + 5897*x^10 - 12483*x^9 + 31044*x^8 - 47785*x^7 + 91545*x^6 - 91886*x^5 + 136834*x^4 - 79970*x^3 + 96634*x^2 - 49215*x + 85039, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 38 x^{14} - 130 x^{13} + 641 x^{12} - 1757 x^{11} + 5897 x^{10} - 12483 x^{9} + 31044 x^{8} - 47785 x^{7} + 91545 x^{6} - 91886 x^{5} + 136834 x^{4} - 79970 x^{3} + 96634 x^{2} - 49215 x + 85039 \)

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:  \(2569453321037674837093389=3^{10}\cdot 61^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.54$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{34515560325784499630027734247120551} a^{15} + \frac{5821280831974452921734694042193750}{34515560325784499630027734247120551} a^{14} - \frac{13443076130736492010280738572360211}{34515560325784499630027734247120551} a^{13} + \frac{6178672956150397272284559239152078}{34515560325784499630027734247120551} a^{12} + \frac{12434658858349680433202441318317020}{34515560325784499630027734247120551} a^{11} - \frac{9935084274029835665567172390233656}{34515560325784499630027734247120551} a^{10} - \frac{9878010947954180467641957101156998}{34515560325784499630027734247120551} a^{9} - \frac{12626987282361013080479921585441443}{34515560325784499630027734247120551} a^{8} - \frac{82958320477997128342407719226866}{34515560325784499630027734247120551} a^{7} - \frac{27552513154274392858476590631008}{34515560325784499630027734247120551} a^{6} - \frac{3665780645704220862366788070440645}{34515560325784499630027734247120551} a^{5} + \frac{10412425782209206921641926788080808}{34515560325784499630027734247120551} a^{4} - \frac{16676176849206518996205890608296205}{34515560325784499630027734247120551} a^{3} + \frac{4964979462268394368346248191465752}{34515560325784499630027734247120551} a^{2} - \frac{16681359139417630686654150632414888}{34515560325784499630027734247120551} a + \frac{5335052482325217792890671017973824}{34515560325784499630027734247120551}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{34387595095099780}{6901354070863061670877} a^{15} - \frac{232327293607026785}{6901354070863061670877} a^{14} + \frac{1415826861757443400}{6901354070863061670877} a^{13} - \frac{6913417440976917510}{6901354070863061670877} a^{12} + \frac{24653663963740767175}{6901354070863061670877} a^{11} - \frac{86796530147784447528}{6901354070863061670877} a^{10} + \frac{216141575917438506770}{6901354070863061670877} a^{9} - \frac{567310675895822987190}{6901354070863061670877} a^{8} + \frac{953703551823596720587}{6901354070863061670877} a^{7} - \frac{1887265927409504101444}{6901354070863061670877} a^{6} + \frac{1560268796062567141682}{6901354070863061670877} a^{5} - \frac{2632517943441198492412}{6901354070863061670877} a^{4} - \frac{1925019492910097756340}{6901354070863061670877} a^{3} - \frac{851210500749777478608}{6901354070863061670877} a^{2} - \frac{4145091270211868892536}{6901354070863061670877} a + \frac{1951142008569674591775}{6901354070863061670877} \) (order $6$)
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:  \( 302344.361334 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.549.1, 8.0.18385461.1

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 $16$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $16$ $16$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$61$61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.2.2$x^{4} - 61 x^{2} + 7442$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
61.8.6.2$x^{8} + 183 x^{4} + 14884$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$