↑

Properties

Label 16.0.22850732699...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 7^{4}\cdot 29^{6}$
Root discriminant $38.45$
Ramified primes $2, 5, 7, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $(C_2^2\times C_4).C_2^4$ (as 16T471)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![521, -4254, 17302, -40390, 55945, -44736, 16641, 2060, -4046, 420, 974, -518, 55, 40, -12, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 12*x^14 + 40*x^13 + 55*x^12 - 518*x^11 + 974*x^10 + 420*x^9 - 4046*x^8 + 2060*x^7 + 16641*x^6 - 44736*x^5 + 55945*x^4 - 40390*x^3 + 17302*x^2 - 4254*x + 521)
 
gp: K = bnfinit(x^16 - 2*x^15 - 12*x^14 + 40*x^13 + 55*x^12 - 518*x^11 + 974*x^10 + 420*x^9 - 4046*x^8 + 2060*x^7 + 16641*x^6 - 44736*x^5 + 55945*x^4 - 40390*x^3 + 17302*x^2 - 4254*x + 521, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 12 x^{14} + 40 x^{13} + 55 x^{12} - 518 x^{11} + 974 x^{10} + 420 x^{9} - 4046 x^{8} + 2060 x^{7} + 16641 x^{6} - 44736 x^{5} + 55945 x^{4} - 40390 x^{3} + 17302 x^{2} - 4254 x + 521 \)

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:  \(22850732699536000000000000=2^{16}\cdot 5^{12}\cdot 7^{4}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.45$
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:  $2, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} - \frac{3}{13} a^{13} + \frac{5}{13} a^{12} + \frac{6}{13} a^{11} + \frac{2}{13} a^{10} + \frac{6}{13} a^{9} - \frac{5}{13} a^{8} + \frac{2}{13} a^{7} + \frac{3}{13} a^{6} + \frac{5}{13} a^{5} - \frac{1}{13} a^{4} + \frac{3}{13} a^{3} + \frac{2}{13} a^{2} + \frac{2}{13} a - \frac{1}{13}$, $\frac{1}{173381281994629007} a^{15} - \frac{3880938670183020}{173381281994629007} a^{14} - \frac{41322804247474131}{173381281994629007} a^{13} - \frac{50160802684199206}{173381281994629007} a^{12} + \frac{74650982798473610}{173381281994629007} a^{11} + \frac{24517457773549667}{173381281994629007} a^{10} + \frac{63758680784798703}{173381281994629007} a^{9} + \frac{3428102502148292}{173381281994629007} a^{8} + \frac{51398468345733289}{173381281994629007} a^{7} + \frac{40337938198089950}{173381281994629007} a^{6} + \frac{31007839336440701}{173381281994629007} a^{5} + \frac{36326784070578317}{173381281994629007} a^{4} + \frac{34010826386795803}{173381281994629007} a^{3} - \frac{5833735709417463}{13337021691894539} a^{2} + \frac{52717324387119220}{173381281994629007} a - \frac{30615488628427715}{173381281994629007}$

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

Class group and class number

$C_{4}$, 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:  \( \frac{168541714066926}{173381281994629007} a^{15} - \frac{10485698558724482}{173381281994629007} a^{14} - \frac{82975091764306}{173381281994629007} a^{13} + \frac{146911253812085663}{173381281994629007} a^{12} - \frac{140771585932696436}{173381281994629007} a^{11} - \frac{1112931899663012906}{173381281994629007} a^{10} + \frac{3731094962099495602}{173381281994629007} a^{9} - \frac{1895289853235805510}{173381281994629007} a^{8} - \frac{13259804024869525966}{173381281994629007} a^{7} + \frac{20672666261685957180}{173381281994629007} a^{6} + \frac{36199644895032399698}{173381281994629007} a^{5} - \frac{140886974019608035711}{173381281994629007} a^{4} + \frac{173116135778490013132}{173381281994629007} a^{3} - \frac{102124617894360222021}{173381281994629007} a^{2} + \frac{30078842158264223878}{173381281994629007} a - \frac{354634598788306835}{13337021691894539} \) (order $10$)
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:  \( 969289.203285 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4).C_2^4$ (as 16T471):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $(C_2^2\times C_4).C_2^4$
Character table for $(C_2^2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.11600.1, 4.4.58000.1, 8.0.3364000000.5

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 R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$