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Properties

Label 16.0.26231448756...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{4}\cdot 41^{4}\cdot 97^{4}$
Root discriminant $33.59$
Ramified primes $2, 5, 41, 97$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_2^4.C_2^4$ (as 16T573)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 0, 10752, -1536, 9024, 1344, 2328, 2488, 169, 922, 225, 76, 129, -16, 23, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 23*x^14 - 16*x^13 + 129*x^12 + 76*x^11 + 225*x^10 + 922*x^9 + 169*x^8 + 2488*x^7 + 2328*x^6 + 1344*x^5 + 9024*x^4 - 1536*x^3 + 10752*x^2 + 4096)
 
gp: K = bnfinit(x^16 - 2*x^15 + 23*x^14 - 16*x^13 + 129*x^12 + 76*x^11 + 225*x^10 + 922*x^9 + 169*x^8 + 2488*x^7 + 2328*x^6 + 1344*x^5 + 9024*x^4 - 1536*x^3 + 10752*x^2 + 4096, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 23 x^{14} - 16 x^{13} + 129 x^{12} + 76 x^{11} + 225 x^{10} + 922 x^{9} + 169 x^{8} + 2488 x^{7} + 2328 x^{6} + 1344 x^{5} + 9024 x^{4} - 1536 x^{3} + 10752 x^{2} + 4096 \)

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:  \(2623144875606314844160000=2^{24}\cdot 5^{4}\cdot 41^{4}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.59$
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:  $2, 5, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{16} a^{11} + \frac{3}{16} a^{9} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{7}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{1}{64} a^{10} + \frac{25}{64} a^{8} - \frac{5}{16} a^{7} + \frac{9}{64} a^{6} + \frac{13}{32} a^{5} + \frac{17}{64} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{128} a^{13} + \frac{3}{128} a^{11} - \frac{1}{64} a^{10} - \frac{15}{128} a^{9} + \frac{7}{64} a^{8} - \frac{23}{128} a^{7} - \frac{9}{32} a^{6} - \frac{51}{128} a^{5} + \frac{21}{64} a^{4} - \frac{1}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{20992} a^{14} + \frac{25}{10496} a^{13} + \frac{47}{20992} a^{12} - \frac{105}{5248} a^{11} - \frac{1087}{20992} a^{10} - \frac{49}{328} a^{9} - \frac{2479}{20992} a^{8} + \frac{2439}{10496} a^{7} + \frac{1009}{20992} a^{6} - \frac{2509}{5248} a^{5} + \frac{479}{2624} a^{4} + \frac{71}{656} a^{3} - \frac{141}{328} a^{2} + \frac{11}{41} a - \frac{7}{41}$, $\frac{1}{6690932871887872} a^{15} - \frac{1027484513}{104545826123248} a^{14} - \frac{8026868107949}{6690932871887872} a^{13} + \frac{23441774965295}{3345466435943936} a^{12} - \frac{98216197751135}{6690932871887872} a^{11} + \frac{119448988100167}{3345466435943936} a^{10} + \frac{37137515802425}{6690932871887872} a^{9} + \frac{141856494403751}{1672733217971968} a^{8} + \frac{1924086672645661}{6690932871887872} a^{7} + \frac{326275025088869}{3345466435943936} a^{6} + \frac{286911488662217}{836366608985984} a^{5} - \frac{126668651431009}{418183304492992} a^{4} - \frac{48743183849169}{104545826123248} a^{3} + \frac{23965982104359}{52272913061624} a^{2} + \frac{2719231859465}{6534114132703} a - \frac{2631889605197}{6534114132703}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( 45044.1596417 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^4$ (as 16T573):

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

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 4.0.6208.2, 4.0.254528.2, 8.4.172134400.1, 8.0.64784502784.2, 8.4.1619612569600.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$