↑

Properties

Label 16.0.89060441849...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{52}\cdot 3^{4}\cdot 5^{12}$
Root discriminant $41.87$
Ramified primes $2, 3, 5$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
Galois group $C_4^2:C_2$ (as 16T17)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4241, 11976, 20608, 12000, 358, -9336, -1704, 2384, 3177, -352, -848, -24, 174, -8, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 - 8*x^13 + 174*x^12 - 24*x^11 - 848*x^10 - 352*x^9 + 3177*x^8 + 2384*x^7 - 1704*x^6 - 9336*x^5 + 358*x^4 + 12000*x^3 + 20608*x^2 + 11976*x + 4241)
 
gp: K = bnfinit(x^16 - 16*x^14 - 8*x^13 + 174*x^12 - 24*x^11 - 848*x^10 - 352*x^9 + 3177*x^8 + 2384*x^7 - 1704*x^6 - 9336*x^5 + 358*x^4 + 12000*x^3 + 20608*x^2 + 11976*x + 4241, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} - 8 x^{13} + 174 x^{12} - 24 x^{11} - 848 x^{10} - 352 x^{9} + 3177 x^{8} + 2384 x^{7} - 1704 x^{6} - 9336 x^{5} + 358 x^{4} + 12000 x^{3} + 20608 x^{2} + 11976 x + 4241 \)

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:  \(89060441849856000000000000=2^{52}\cdot 3^{4}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.87$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{172535790196870436355874617964683} a^{15} + \frac{2010912446581911829304240116157}{172535790196870436355874617964683} a^{14} - \frac{2153551470201193962379339878310}{57511930065623478785291539321561} a^{13} - \frac{31238490337654136692018964972876}{172535790196870436355874617964683} a^{12} - \frac{11477639277465304893641378005159}{172535790196870436355874617964683} a^{11} - \frac{86193760086437509843211930512622}{172535790196870436355874617964683} a^{10} + \frac{27672220063338073220867702824912}{57511930065623478785291539321561} a^{9} - \frac{2011740700613679780735515365105}{172535790196870436355874617964683} a^{8} - \frac{42631935891017645543666615489036}{172535790196870436355874617964683} a^{7} - \frac{19363763138152513682885021133350}{172535790196870436355874617964683} a^{6} + \frac{55755812004928925607458768866184}{172535790196870436355874617964683} a^{5} + \frac{83156572954058885629728443755960}{172535790196870436355874617964683} a^{4} + \frac{2403607103884961789094970160952}{57511930065623478785291539321561} a^{3} + \frac{20520144835645473593738335388274}{57511930065623478785291539321561} a^{2} + \frac{13698555921255920589857789486647}{172535790196870436355874617964683} a + \frac{58577749479210825368674462068302}{172535790196870436355874617964683}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( -1 \) (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:  \( 85299.4255313 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4):C_4$ (as 16T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4^2:C_2$
Character table for $C_4^2:C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.256000.2, 4.4.256000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.65536000000.1, 8.0.589824000000.12, 8.0.5898240000.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed