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Properties

Label 16.16.1144539176...4321.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{12}\cdot 53^{12}$
Root discriminant $134.48$
Ramified primes $13, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8161, -1963061, -1439221, 30604631, -58857565, 33312812, 11515707, -18037073, 4184332, 1107152, -425769, -23010, 14729, 188, -212, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 212*x^14 + 188*x^13 + 14729*x^12 - 23010*x^11 - 425769*x^10 + 1107152*x^9 + 4184332*x^8 - 18037073*x^7 + 11515707*x^6 + 33312812*x^5 - 58857565*x^4 + 30604631*x^3 - 1439221*x^2 - 1963061*x - 8161)
 
gp: K = bnfinit(x^16 - x^15 - 212*x^14 + 188*x^13 + 14729*x^12 - 23010*x^11 - 425769*x^10 + 1107152*x^9 + 4184332*x^8 - 18037073*x^7 + 11515707*x^6 + 33312812*x^5 - 58857565*x^4 + 30604631*x^3 - 1439221*x^2 - 1963061*x - 8161, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 212 x^{14} + 188 x^{13} + 14729 x^{12} - 23010 x^{11} - 425769 x^{10} + 1107152 x^{9} + 4184332 x^{8} - 18037073 x^{7} + 11515707 x^{6} + 33312812 x^{5} - 58857565 x^{4} + 30604631 x^{3} - 1439221 x^{2} - 1963061 x - 8161 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11445391768549949624817238631584321=13^{12}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.48$
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 Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{3} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{2958} a^{14} - \frac{71}{1479} a^{13} + \frac{36}{493} a^{12} + \frac{647}{2958} a^{11} + \frac{3}{493} a^{10} - \frac{38}{493} a^{9} - \frac{695}{1479} a^{8} - \frac{977}{2958} a^{7} - \frac{167}{2958} a^{6} + \frac{445}{986} a^{5} - \frac{175}{986} a^{4} - \frac{499}{1479} a^{3} + \frac{117}{986} a^{2} - \frac{90}{493} a + \frac{413}{2958}$, $\frac{1}{16139583018838645628088999422012509400116011105486} a^{15} - \frac{2031070090788366997953025413627422417003629733}{16139583018838645628088999422012509400116011105486} a^{14} + \frac{37547411591020816676547129807186669658455819020}{2689930503139774271348166570335418233352668517581} a^{13} - \frac{74925902115947435073256919755087165590986887621}{2689930503139774271348166570335418233352668517581} a^{12} + \frac{331333074459142009445815309705524854670998290880}{2689930503139774271348166570335418233352668517581} a^{11} - \frac{89576211901690354726170285508322399995487923099}{375339139972991758792767428418895567444558397802} a^{10} - \frac{537493371118043673378621644218854234555760134935}{16139583018838645628088999422012509400116011105486} a^{9} + \frac{5688445319031766013848048629983289665082937360495}{16139583018838645628088999422012509400116011105486} a^{8} + \frac{1946413705794749735044885207700352366836815883513}{5379861006279548542696333140670836466705337035162} a^{7} + \frac{370937691287495259172302923727245745861472681475}{2689930503139774271348166570335418233352668517581} a^{6} - \frac{3336914757394654838390844328652767541912265240697}{16139583018838645628088999422012509400116011105486} a^{5} - \frac{3594558587910229713405313828812148864162418928305}{8069791509419322814044499711006254700058005552743} a^{4} - \frac{7739232903443709896311257319058770470722424180479}{16139583018838645628088999422012509400116011105486} a^{3} - \frac{89396602149606945320288705745383690242857119003}{5379861006279548542696333140670836466705337035162} a^{2} - \frac{3355471587143347502860862640889536844924809201253}{8069791509419322814044499711006254700058005552743} a - \frac{5320832694542607506318388330424445479319268378371}{16139583018838645628088999422012509400116011105486}$

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:  $15$
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:  \( 378472370764 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{689}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{53})\), 4.4.36517.1 x2, 4.4.8957.1 x2, 4.4.327082769.1, 4.4.327082769.2, 8.8.225360027841.1, 8.8.106983137776707361.2, 8.8.106983137776707361.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ 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$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$