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Properties

Label 16.4.23126068311...5856.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 3^{12}\cdot 11^{10}$
Root discriminant $28.86$
Ramified primes $2, 3, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T860

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -144, -1032, -1896, -1196, -1344, -1416, -276, -438, -72, -72, -54, 7, -6, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^14 - 6*x^13 + 7*x^12 - 54*x^11 - 72*x^10 - 72*x^9 - 438*x^8 - 276*x^7 - 1416*x^6 - 1344*x^5 - 1196*x^4 - 1896*x^3 - 1032*x^2 - 144*x + 16)
 
gp: K = bnfinit(x^16 + 6*x^14 - 6*x^13 + 7*x^12 - 54*x^11 - 72*x^10 - 72*x^9 - 438*x^8 - 276*x^7 - 1416*x^6 - 1344*x^5 - 1196*x^4 - 1896*x^3 - 1032*x^2 - 144*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} + 6 x^{14} - 6 x^{13} + 7 x^{12} - 54 x^{11} - 72 x^{10} - 72 x^{9} - 438 x^{8} - 276 x^{7} - 1416 x^{6} - 1344 x^{5} - 1196 x^{4} - 1896 x^{3} - 1032 x^{2} - 144 x + 16 \)

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:  \(231260683111582301945856=2^{24}\cdot 3^{12}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.86$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{44} a^{12} + \frac{3}{22} a^{11} - \frac{5}{22} a^{10} + \frac{1}{11} a^{9} - \frac{9}{44} a^{8} + \frac{1}{11} a^{7} - \frac{1}{22} a^{6} - \frac{1}{22} a^{5} - \frac{3}{11} a^{4} - \frac{2}{11} a^{3} + \frac{2}{11} a^{2} + \frac{5}{11} a + \frac{5}{11}$, $\frac{1}{44} a^{13} - \frac{1}{22} a^{11} - \frac{1}{22} a^{10} - \frac{1}{4} a^{9} - \frac{2}{11} a^{8} - \frac{1}{11} a^{7} + \frac{5}{22} a^{6} + \frac{5}{11} a^{4} + \frac{3}{11} a^{3} + \frac{4}{11} a^{2} - \frac{3}{11} a + \frac{3}{11}$, $\frac{1}{88} a^{14} + \frac{5}{44} a^{11} - \frac{9}{88} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{22} a^{7} + \frac{9}{44} a^{6} + \frac{2}{11} a^{5} - \frac{3}{22} a^{4} + \frac{1}{22} a^{2} + \frac{1}{11} a + \frac{5}{11}$, $\frac{1}{579039082019219272} a^{15} - \frac{109735809999931}{26319958273600876} a^{14} + \frac{212050023634554}{72379885252402409} a^{13} - \frac{712089430662771}{72379885252402409} a^{12} + \frac{103461993416136371}{579039082019219272} a^{11} - \frac{10110212400497177}{72379885252402409} a^{10} - \frac{5546901148821933}{26319958273600876} a^{9} + \frac{17923774982550617}{289519541009609636} a^{8} + \frac{25018194987220095}{289519541009609636} a^{7} + \frac{23381251856241535}{144759770504804818} a^{6} + \frac{63089094820100533}{144759770504804818} a^{5} - \frac{51239962347633655}{144759770504804818} a^{4} - \frac{46996128866620459}{144759770504804818} a^{3} + \frac{13178130210968237}{72379885252402409} a^{2} - \frac{26730489379530938}{72379885252402409} a + \frac{12107435069252100}{72379885252402409}$

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:  \( 225898.327942 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T860:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n860 are not computed
Character table for t16n860 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{11}) \), 4.4.4752.1 x2, 4.4.13068.1 x2, \(\Q(\sqrt{3}, \sqrt{11})\), 8.8.2732361984.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.16.22$x^{8} + 4 x^{6} + 4 x^{4} + 16$$4$$2$$16$$Q_8:C_2$$[2, 3, 3]^{2}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$