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Properties

Label 16.8.24109722907...7601.7
Degree $16$
Signature $[8, 4]$
Discriminant $13^{10}\cdot 53^{10}$
Root discriminant $59.41$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1223, 7020, 25989, 80707, 17648, -196260, 41440, 57243, -20627, 4906, -3039, 397, 1, 55, -7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 7*x^14 + 55*x^13 + x^12 + 397*x^11 - 3039*x^10 + 4906*x^9 - 20627*x^8 + 57243*x^7 + 41440*x^6 - 196260*x^5 + 17648*x^4 + 80707*x^3 + 25989*x^2 + 7020*x - 1223)
 
gp: K = bnfinit(x^16 - 4*x^15 - 7*x^14 + 55*x^13 + x^12 + 397*x^11 - 3039*x^10 + 4906*x^9 - 20627*x^8 + 57243*x^7 + 41440*x^6 - 196260*x^5 + 17648*x^4 + 80707*x^3 + 25989*x^2 + 7020*x - 1223, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 7 x^{14} + 55 x^{13} + x^{12} + 397 x^{11} - 3039 x^{10} + 4906 x^{9} - 20627 x^{8} + 57243 x^{7} + 41440 x^{6} - 196260 x^{5} + 17648 x^{4} + 80707 x^{3} + 25989 x^{2} + 7020 x - 1223 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{39} a^{14} + \frac{2}{39} a^{13} + \frac{3}{13} a^{12} - \frac{2}{39} a^{10} - \frac{5}{39} a^{9} + \frac{4}{39} a^{8} - \frac{4}{39} a^{7} - \frac{4}{39} a^{6} - \frac{10}{39} a^{5} - \frac{5}{13} a^{4} + \frac{14}{39} a^{3} + \frac{5}{39} a^{2} - \frac{5}{13} a - \frac{16}{39}$, $\frac{1}{6991744949363697118211564246525322898803} a^{15} - \frac{41176651697022037480589215755752826953}{6991744949363697118211564246525322898803} a^{14} - \frac{185570893114476031436771829560378336405}{6991744949363697118211564246525322898803} a^{13} - \frac{789459788200783674266499948784921649386}{2330581649787899039403854748841774299601} a^{12} - \frac{259762812391686043160144591551456893602}{6991744949363697118211564246525322898803} a^{11} - \frac{992485091745095033384814965525818511491}{2330581649787899039403854748841774299601} a^{10} + \frac{792543508778716160240353634623775310914}{2330581649787899039403854748841774299601} a^{9} - \frac{2728960306087264784073118543585921738301}{6991744949363697118211564246525322898803} a^{8} + \frac{463933081851861271587881423054768668253}{2330581649787899039403854748841774299601} a^{7} + \frac{1085860422107077712609513156066894813852}{2330581649787899039403854748841774299601} a^{6} - \frac{3373967732918759056531025087742456918437}{6991744949363697118211564246525322898803} a^{5} - \frac{2794464581258993410859920531637800991494}{6991744949363697118211564246525322898803} a^{4} - \frac{106847462850305424195793269947431899606}{2330581649787899039403854748841774299601} a^{3} - \frac{180348711643227886504970866091831938259}{537826534566438239862428018963486376831} a^{2} - \frac{2849764425982372506418130773156490844352}{6991744949363697118211564246525322898803} a - \frac{3376058593693230463686043043710519805012}{6991744949363697118211564246525322898803}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 195139051.528 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{689}) \), 4.4.8957.1 x2, 4.4.36517.1 x2, \(\Q(\sqrt{13}, \sqrt{53})\), 8.8.225360027841.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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