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Properties

Label 16.0.20723941878...0625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{8}\cdot 101^{4}$
Root discriminant $38.22$
Ramified primes $5, 13, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12401, -14598, -6502, 17395, -10935, -5082, 12899, -12185, 9704, -4745, 2716, -754, 405, -50, 32, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 32*x^14 - 50*x^13 + 405*x^12 - 754*x^11 + 2716*x^10 - 4745*x^9 + 9704*x^8 - 12185*x^7 + 12899*x^6 - 5082*x^5 - 10935*x^4 + 17395*x^3 - 6502*x^2 - 14598*x + 12401)
 
gp: K = bnfinit(x^16 - x^15 + 32*x^14 - 50*x^13 + 405*x^12 - 754*x^11 + 2716*x^10 - 4745*x^9 + 9704*x^8 - 12185*x^7 + 12899*x^6 - 5082*x^5 - 10935*x^4 + 17395*x^3 - 6502*x^2 - 14598*x + 12401, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 32 x^{14} - 50 x^{13} + 405 x^{12} - 754 x^{11} + 2716 x^{10} - 4745 x^{9} + 9704 x^{8} - 12185 x^{7} + 12899 x^{6} - 5082 x^{5} - 10935 x^{4} + 17395 x^{3} - 6502 x^{2} - 14598 x + 12401 \)

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:  \(20723941878730254150390625=5^{12}\cdot 13^{8}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.22$
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:  $5, 13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1291} a^{14} + \frac{578}{1291} a^{13} - \frac{207}{1291} a^{12} - \frac{78}{1291} a^{11} - \frac{474}{1291} a^{10} - \frac{33}{1291} a^{9} - \frac{476}{1291} a^{8} + \frac{571}{1291} a^{7} - \frac{315}{1291} a^{6} - \frac{16}{1291} a^{5} - \frac{488}{1291} a^{4} - \frac{266}{1291} a^{3} + \frac{424}{1291} a^{2} + \frac{320}{1291} a - \frac{314}{1291}$, $\frac{1}{90075890583554831894591814731} a^{15} - \frac{29176971099481400494133898}{90075890583554831894591814731} a^{14} + \frac{31246610737513491186351436251}{90075890583554831894591814731} a^{13} + \frac{7839943108320074898680353209}{90075890583554831894591814731} a^{12} - \frac{22774974423800920705044464839}{90075890583554831894591814731} a^{11} - \frac{15392666880135763290248500218}{90075890583554831894591814731} a^{10} - \frac{15661471427895172604391931442}{90075890583554831894591814731} a^{9} - \frac{41991361540168077104581055356}{90075890583554831894591814731} a^{8} + \frac{17830736087343107491275279691}{90075890583554831894591814731} a^{7} - \frac{42995944178918401623453215849}{90075890583554831894591814731} a^{6} + \frac{6257620980469622579932354041}{90075890583554831894591814731} a^{5} - \frac{8125771116799255803841271932}{90075890583554831894591814731} a^{4} + \frac{10867934708271573633939800956}{90075890583554831894591814731} a^{3} + \frac{38641424779048752811654114666}{90075890583554831894591814731} a^{2} + \frac{43429791402315497083229853096}{90075890583554831894591814731} a + \frac{23197180304123880609316688391}{90075890583554831894591814731}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{3539245561949033493371}{136272149143048157177899871} a^{15} + \frac{5333378958653388461793}{136272149143048157177899871} a^{14} + \frac{120915608831625268117679}{136272149143048157177899871} a^{13} + \frac{97618647931771672283120}{136272149143048157177899871} a^{12} + \frac{1546572677053332257798318}{136272149143048157177899871} a^{11} + \frac{398669883761793497447120}{136272149143048157177899871} a^{10} + \frac{10264837762071907608013458}{136272149143048157177899871} a^{9} - \frac{914078380268184697394725}{136272149143048157177899871} a^{8} + \frac{40730278370384440750333114}{136272149143048157177899871} a^{7} - \frac{8470702076012721832246014}{136272149143048157177899871} a^{6} + \frac{92856866898669172571982026}{136272149143048157177899871} a^{5} - \frac{16470700573217496565539567}{136272149143048157177899871} a^{4} + \frac{78004526584220637282938814}{136272149143048157177899871} a^{3} - \frac{9671081204120478669051734}{136272149143048157177899871} a^{2} - \frac{5066518556442734882016219}{136272149143048157177899871} a - \frac{23144933778064009838091989}{136272149143048157177899871} \) (order $10$)
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:  \( 727465.154558 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), 4.0.21125.1, \(\Q(\zeta_{5})\), \(\Q(\sqrt{5}, \sqrt{13})\), 8.0.446265625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$