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Properties

Label 16.4.24109722907...601.26
Degree $16$
Signature $[4, 6]$
Discriminant $13^{10}\cdot 53^{10}$
Root discriminant $59.41$
Ramified primes $13, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2928, -10384, 26008, -24142, -25355, 90186, -100244, 58816, -18360, 1398, 1929, -1053, 74, 94, -21, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 21*x^14 + 94*x^13 + 74*x^12 - 1053*x^11 + 1929*x^10 + 1398*x^9 - 18360*x^8 + 58816*x^7 - 100244*x^6 + 90186*x^5 - 25355*x^4 - 24142*x^3 + 26008*x^2 - 10384*x + 2928)
 
gp: K = bnfinit(x^16 - 3*x^15 - 21*x^14 + 94*x^13 + 74*x^12 - 1053*x^11 + 1929*x^10 + 1398*x^9 - 18360*x^8 + 58816*x^7 - 100244*x^6 + 90186*x^5 - 25355*x^4 - 24142*x^3 + 26008*x^2 - 10384*x + 2928, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 21 x^{14} + 94 x^{13} + 74 x^{12} - 1053 x^{11} + 1929 x^{10} + 1398 x^{9} - 18360 x^{8} + 58816 x^{7} - 100244 x^{6} + 90186 x^{5} - 25355 x^{4} - 24142 x^{3} + 26008 x^{2} - 10384 x + 2928 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{13} + \frac{1}{24} a^{12} + \frac{1}{6} a^{11} + \frac{1}{8} a^{10} + \frac{1}{4} a^{9} - \frac{1}{12} a^{8} + \frac{7}{24} a^{7} + \frac{1}{6} a^{6} + \frac{11}{24} a^{5} - \frac{1}{3} a^{4} - \frac{3}{8} a^{3} - \frac{5}{12} a^{2} + \frac{1}{6} a$, $\frac{1}{24} a^{14} - \frac{1}{8} a^{12} + \frac{5}{24} a^{11} + \frac{1}{8} a^{10} - \frac{1}{12} a^{9} + \frac{3}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{24} a^{6} + \frac{5}{24} a^{5} + \frac{5}{24} a^{4} + \frac{11}{24} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{13260812758899226480557209777328} a^{15} - \frac{16333396656986014568622660568}{828800797431201655034825611083} a^{14} + \frac{103556871635187152732985655915}{13260812758899226480557209777328} a^{13} - \frac{540171649085484824303177897435}{4420270919633075493519069925776} a^{12} + \frac{1666357011824785883945525963987}{13260812758899226480557209777328} a^{11} - \frac{28514313079130646383070336919}{1657601594862403310069651222166} a^{10} - \frac{4958182746381394911307396135951}{13260812758899226480557209777328} a^{9} + \frac{2266411644765021048472447558525}{13260812758899226480557209777328} a^{8} - \frac{520061352296512074676141507225}{13260812758899226480557209777328} a^{7} + \frac{3605112880186525150518590556461}{13260812758899226480557209777328} a^{6} + \frac{1687331284087009139225045842391}{13260812758899226480557209777328} a^{5} - \frac{1043786643102179239746063553745}{13260812758899226480557209777328} a^{4} - \frac{995990247713790385183780304807}{2210135459816537746759534962888} a^{3} + \frac{762891987341657488418435087597}{3315203189724806620139302444332} a^{2} + \frac{312214410369448778593198932133}{1657601594862403310069651222166} a - \frac{61292342883944169350732077532}{276266932477067218344941870361}$

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:  $9$
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:  \( 126898100.761 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.225360027841.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.2.1$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}$
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}$
53Data not computed