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Properties

Label 16.0.63291260341...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{12}\cdot 11^{6}\cdot 151^{4}$
Root discriminant $230.45$
Ramified primes $2, 5, 11, 151$
Class number $18230400$ (GRH)
Class group $[2, 4, 4, 569700]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1006498523536, 0, 533244250880, 0, 104285207312, 0, 9477533120, 0, 431415164, 0, 9846560, 0, 109988, 0, 560, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 560*x^14 + 109988*x^12 + 9846560*x^10 + 431415164*x^8 + 9477533120*x^6 + 104285207312*x^4 + 533244250880*x^2 + 1006498523536)
 
gp: K = bnfinit(x^16 + 560*x^14 + 109988*x^12 + 9846560*x^10 + 431415164*x^8 + 9477533120*x^6 + 104285207312*x^4 + 533244250880*x^2 + 1006498523536, 1)
 

Normalized defining polynomial

\( x^{16} + 560 x^{14} + 109988 x^{12} + 9846560 x^{10} + 431415164 x^{8} + 9477533120 x^{6} + 104285207312 x^{4} + 533244250880 x^{2} + 1006498523536 \)

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:  \(63291260341991767325802496000000000000=2^{48}\cdot 5^{12}\cdot 11^{6}\cdot 151^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $230.45$
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:  $2, 5, 11, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{56} a^{8} + \frac{1}{28} a^{4} - \frac{2}{7} a^{2} + \frac{1}{14}$, $\frac{1}{112} a^{9} - \frac{1}{4} a^{7} - \frac{13}{56} a^{5} - \frac{1}{7} a^{3} + \frac{1}{28} a$, $\frac{1}{16912} a^{10} - \frac{11}{4228} a^{8} - \frac{1329}{8456} a^{6} - \frac{66}{1057} a^{4} + \frac{1437}{4228} a^{2} - \frac{1}{7}$, $\frac{1}{16912} a^{11} - \frac{11}{4228} a^{9} - \frac{1329}{8456} a^{7} - \frac{66}{1057} a^{5} + \frac{1437}{4228} a^{3} - \frac{1}{7} a$, $\frac{1}{28090832} a^{12} + \frac{5}{250811} a^{10} + \frac{27497}{7022708} a^{8} - \frac{23235}{3511354} a^{6} - \frac{624497}{3511354} a^{4} - \frac{344}{1057} a^{2} + \frac{1}{14}$, $\frac{1}{56181664} a^{13} + \frac{5}{501622} a^{11} + \frac{27497}{14045416} a^{9} + \frac{866221}{3511354} a^{7} - \frac{624497}{7022708} a^{5} + \frac{713}{2114} a^{3} + \frac{1}{28} a$, $\frac{1}{2331012845295188201698815264} a^{14} - \frac{2484183626132929921}{166500917521084871549915376} a^{12} + \frac{2827569822754021989353}{291376605661898525212351908} a^{10} + \frac{5011370739759557738782237}{582753211323797050424703816} a^{8} - \frac{911782783780515181417229}{145688302830949262606175954} a^{6} - \frac{100402887761441994106621}{1929646395111910762995708} a^{4} + \frac{315296562124123273975}{1161737745401511597228} a^{2} + \frac{676802597086232893}{1923406863247535757}$, $\frac{1}{2331012845295188201698815264} a^{15} + \frac{6712062998478681007}{2331012845295188201698815264} a^{13} + \frac{5731914186257800982423}{291376605661898525212351908} a^{11} + \frac{1898170047687408462050357}{1165506422647594100849407632} a^{9} - \frac{398228634725066184290599}{41625229380271217887478844} a^{7} + \frac{31992035372708173856687}{350844799111256502362856} a^{5} - \frac{288653192935602953723}{1161737745401511597228} a^{3} + \frac{676802597086232893}{1923406863247535757} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{569700}$, which has order $18230400$ (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:  \( 126260.424946 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

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 $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.22000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.88000.1, 8.8.123904000000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$151$151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.4.2.1$x^{4} + 3473 x^{2} + 3283344$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
151.4.2.1$x^{4} + 3473 x^{2} + 3283344$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$