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Properties

Label 16.0.22825217147...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{12}\cdot 5^{8}$
Root discriminant $28.83$
Ramified primes $2, 3, 5$
Class number $8$
Class group $[2, 2, 2]$
Galois group $Q_8 : C_2$ (as 16T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1156, 816, -1312, -2288, 3188, -2712, 3056, -1344, 874, -576, 280, -112, 78, -16, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 - 16*x^13 + 78*x^12 - 112*x^11 + 280*x^10 - 576*x^9 + 874*x^8 - 1344*x^7 + 3056*x^6 - 2712*x^5 + 3188*x^4 - 2288*x^3 - 1312*x^2 + 816*x + 1156)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 - 16*x^13 + 78*x^12 - 112*x^11 + 280*x^10 - 576*x^9 + 874*x^8 - 1344*x^7 + 3056*x^6 - 2712*x^5 + 3188*x^4 - 2288*x^3 - 1312*x^2 + 816*x + 1156, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} - 16 x^{13} + 78 x^{12} - 112 x^{11} + 280 x^{10} - 576 x^{9} + 874 x^{8} - 1344 x^{7} + 3056 x^{6} - 2712 x^{5} + 3188 x^{4} - 2288 x^{3} - 1312 x^{2} + 816 x + 1156 \)

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:  \(228252171475353600000000=2^{40}\cdot 3^{12}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.83$
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 Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{42} a^{12} - \frac{1}{7} a^{11} + \frac{2}{21} a^{10} - \frac{5}{42} a^{9} + \frac{1}{14} a^{8} + \frac{2}{7} a^{7} + \frac{5}{21} a^{6} - \frac{1}{7} a^{5} + \frac{5}{21} a^{4} + \frac{4}{21} a^{3} - \frac{3}{7} a^{2} + \frac{2}{21} a + \frac{5}{21}$, $\frac{1}{42} a^{13} + \frac{5}{21} a^{11} - \frac{1}{21} a^{10} - \frac{1}{7} a^{9} + \frac{3}{14} a^{8} - \frac{1}{21} a^{7} + \frac{2}{7} a^{6} + \frac{8}{21} a^{5} - \frac{8}{21} a^{4} - \frac{2}{7} a^{3} - \frac{10}{21} a^{2} - \frac{4}{21} a + \frac{3}{7}$, $\frac{1}{16422} a^{14} + \frac{86}{8211} a^{13} + \frac{25}{8211} a^{12} + \frac{86}{357} a^{11} - \frac{452}{8211} a^{10} - \frac{1643}{16422} a^{9} + \frac{8}{1173} a^{8} - \frac{2740}{8211} a^{7} + \frac{3802}{8211} a^{6} + \frac{206}{2737} a^{5} - \frac{31}{161} a^{4} + \frac{52}{391} a^{3} - \frac{2819}{8211} a^{2} - \frac{2027}{8211} a - \frac{164}{483}$, $\frac{1}{316680736638041003382} a^{15} - \frac{6075532837041209}{316680736638041003382} a^{14} - \frac{2103376346245476455}{316680736638041003382} a^{13} + \frac{574166398087442133}{105560245546013667794} a^{12} + \frac{553036373499941258}{158340368319020501691} a^{11} + \frac{41519998464531055255}{316680736638041003382} a^{10} - \frac{6991405018905099701}{28789157876185545762} a^{9} + \frac{1906057814255110082}{22620052617002928813} a^{8} - \frac{45207497685030500968}{158340368319020501691} a^{7} - \frac{3508351746083458781}{9314139312883558923} a^{6} + \frac{78797309027271844816}{158340368319020501691} a^{5} + \frac{50466635459541936295}{158340368319020501691} a^{4} - \frac{778933723126066439}{14394578938092772881} a^{3} + \frac{627389710852651129}{14394578938092772881} a^{2} - \frac{47522105450982054856}{158340368319020501691} a + \frac{530271745209900352}{9314139312883558923}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15197.4244561 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_2$ (as 16T11):

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 $Q_8 : C_2$
Character table for $Q_8 : C_2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{15})\), 8.8.3317760000.1, 8.0.29859840000.1 x2, 8.0.19110297600.3 x2, 8.0.477757440000.4 x2

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

Sibling fields

Degree 8 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 R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$