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Properties

Label 16.4.25605559374...1477.2
Degree $16$
Signature $[4, 6]$
Discriminant $13^{15}\cdot 29^{8}$
Root discriminant $59.64$
Ramified primes $13, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65367, 186813, 113400, -37086, -2275, 43277, -58240, 32487, -15002, 4121, -949, 91, 26, -25, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 25*x^13 + 26*x^12 + 91*x^11 - 949*x^10 + 4121*x^9 - 15002*x^8 + 32487*x^7 - 58240*x^6 + 43277*x^5 - 2275*x^4 - 37086*x^3 + 113400*x^2 + 186813*x + 65367)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^14 - 25*x^13 + 26*x^12 + 91*x^11 - 949*x^10 + 4121*x^9 - 15002*x^8 + 32487*x^7 - 58240*x^6 + 43277*x^5 - 2275*x^4 - 37086*x^3 + 113400*x^2 + 186813*x + 65367, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{14} - 25 x^{13} + 26 x^{12} + 91 x^{11} - 949 x^{10} + 4121 x^{9} - 15002 x^{8} + 32487 x^{7} - 58240 x^{6} + 43277 x^{5} - 2275 x^{4} - 37086 x^{3} + 113400 x^{2} + 186813 x + 65367 \)

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:  \(25605559374504409818155101477=13^{15}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.64$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{36} a^{11} + \frac{1}{18} a^{10} - \frac{1}{36} a^{9} + \frac{1}{36} a^{8} + \frac{11}{36} a^{6} - \frac{2}{9} a^{5} - \frac{5}{18} a^{4} - \frac{13}{36} a^{3} + \frac{1}{18} a^{2} - \frac{1}{4}$, $\frac{1}{36} a^{12} + \frac{1}{36} a^{10} - \frac{1}{12} a^{9} - \frac{2}{9} a^{8} - \frac{1}{36} a^{7} + \frac{1}{6} a^{6} - \frac{5}{36} a^{4} - \frac{7}{18} a^{3} + \frac{2}{9} a^{2} - \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{108} a^{13} + \frac{1}{108} a^{11} - \frac{1}{36} a^{10} - \frac{2}{27} a^{9} - \frac{19}{108} a^{8} - \frac{1}{9} a^{7} - \frac{1}{3} a^{6} + \frac{49}{108} a^{5} - \frac{25}{54} a^{4} - \frac{7}{27} a^{3} - \frac{5}{36} a^{2} + \frac{1}{3} a$, $\frac{1}{915732} a^{14} + \frac{961}{457866} a^{13} - \frac{11003}{915732} a^{12} + \frac{2513}{915732} a^{11} - \frac{36955}{457866} a^{10} - \frac{62501}{915732} a^{9} + \frac{91883}{457866} a^{8} - \frac{5716}{25437} a^{7} + \frac{289597}{915732} a^{6} + \frac{11629}{152622} a^{5} + \frac{186041}{457866} a^{4} + \frac{299575}{915732} a^{3} + \frac{51251}{152622} a^{2} + \frac{14675}{50874} a + \frac{4015}{8479}$, $\frac{1}{536454384958437431686976268} a^{15} + \frac{242874252343622239981}{536454384958437431686976268} a^{14} + \frac{153402249816839382112135}{536454384958437431686976268} a^{13} + \frac{1512292815748929610991983}{536454384958437431686976268} a^{12} + \frac{324694305956866147791466}{134113596239609357921744067} a^{11} - \frac{6909655975976420192070143}{89409064159739571947829378} a^{10} - \frac{34921826381114673050395981}{536454384958437431686976268} a^{9} + \frac{21782126617864506019490447}{178818128319479143895658756} a^{8} + \frac{6768746700040391362647688}{134113596239609357921744067} a^{7} + \frac{40000116496687027048312009}{268227192479218715843488134} a^{6} + \frac{1798385434984172359007785}{14901510693289928657971563} a^{5} - \frac{17673512438776004325706385}{268227192479218715843488134} a^{4} + \frac{16256857709935708518414977}{89409064159739571947829378} a^{3} - \frac{13697229550808569699629175}{29803021386579857315943126} a^{2} - \frac{2709544142914753746365859}{6622893641462190514654028} a + \frac{540470477270508042296089}{6622893641462190514654028}$

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:  \( 356190050.114 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.63713.1, 8.4.44380833852277.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $16$

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
13Data not computed
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$