↑

Properties

Label 16.0.89161004482...000.12
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 5^{8}$
Root discriminant $20.39$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $C_8:C_2^2$ (as 16T35)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, -676, 1970, -3632, 4622, -4196, 3030, -1908, 1150, -588, 102, 136, -106, 28, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 28*x^13 - 106*x^12 + 136*x^11 + 102*x^10 - 588*x^9 + 1150*x^8 - 1908*x^7 + 3030*x^6 - 4196*x^5 + 4622*x^4 - 3632*x^3 + 1970*x^2 - 676*x + 169)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 28*x^13 - 106*x^12 + 136*x^11 + 102*x^10 - 588*x^9 + 1150*x^8 - 1908*x^7 + 3030*x^6 - 4196*x^5 + 4622*x^4 - 3632*x^3 + 1970*x^2 - 676*x + 169, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 28 x^{13} - 106 x^{12} + 136 x^{11} + 102 x^{10} - 588 x^{9} + 1150 x^{8} - 1908 x^{7} + 3030 x^{6} - 4196 x^{5} + 4622 x^{4} - 3632 x^{3} + 1970 x^{2} - 676 x + 169 \)

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:  \(891610044825600000000=2^{32}\cdot 3^{12}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.39$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{2} - \frac{3}{10} a + \frac{3}{10}$, $\frac{1}{10} a^{10} + \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{2} a^{2} + \frac{3}{10}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{3}{10}$, $\frac{1}{30} a^{12} + \frac{1}{30} a^{11} + \frac{1}{30} a^{10} - \frac{1}{30} a^{9} + \frac{2}{15} a^{8} + \frac{1}{3} a^{7} + \frac{4}{15} a^{6} - \frac{4}{15} a^{5} + \frac{13}{30} a^{4} + \frac{1}{6} a^{3} + \frac{13}{30} a^{2} + \frac{13}{30} a + \frac{1}{3}$, $\frac{1}{390} a^{13} + \frac{1}{65} a^{12} + \frac{7}{390} a^{10} - \frac{1}{39} a^{9} + \frac{3}{65} a^{8} + \frac{7}{39} a^{7} - \frac{86}{195} a^{6} - \frac{3}{130} a^{5} + \frac{56}{195} a^{4} + \frac{1}{195} a^{3} + \frac{33}{130} a^{2} - \frac{4}{65} a - \frac{1}{3}$, $\frac{1}{390} a^{14} + \frac{1}{130} a^{12} + \frac{7}{390} a^{11} - \frac{1}{30} a^{10} + \frac{1}{390} a^{8} - \frac{23}{195} a^{7} + \frac{3}{130} a^{6} + \frac{44}{195} a^{5} - \frac{17}{78} a^{4} + \frac{29}{130} a^{3} - \frac{37}{130} a^{2} + \frac{7}{195} a - \frac{1}{2}$, $\frac{1}{39037257483510} a^{15} - \frac{3086986439}{13012419161170} a^{14} - \frac{7766113814}{6506209580585} a^{13} - \frac{400213880881}{39037257483510} a^{12} + \frac{222764961561}{6506209580585} a^{11} + \frac{1827283680397}{39037257483510} a^{10} + \frac{215488809566}{6506209580585} a^{9} + \frac{143898584163}{6506209580585} a^{8} - \frac{17359232036747}{39037257483510} a^{7} - \frac{2796565613449}{13012419161170} a^{6} - \frac{2916937026976}{6506209580585} a^{5} - \frac{2260822316671}{7807451496702} a^{4} + \frac{642272990356}{1501432980135} a^{3} + \frac{4979288226293}{13012419161170} a^{2} - \frac{2759549302036}{19518628741755} a + \frac{614996918966}{1501432980135}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -\frac{109104004}{21567545571} a^{15} + \frac{253053373}{21567545571} a^{14} + \frac{426794074}{21567545571} a^{13} - \frac{5983222865}{43135091142} a^{12} + \frac{6193269752}{21567545571} a^{11} + \frac{547185224}{7189181857} a^{10} - \frac{25125451277}{21567545571} a^{9} + \frac{63591464527}{43135091142} a^{8} - \frac{10931726444}{7189181857} a^{7} + \frac{62820357670}{21567545571} a^{6} - \frac{83325512836}{21567545571} a^{5} + \frac{48217007821}{14378363714} a^{4} - \frac{16923945370}{21567545571} a^{3} - \frac{70483134053}{21567545571} a^{2} + \frac{5008488641}{1659041967} a - \frac{3886143671}{3318083934} \) (order $4$)
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:  \( 8516.64407648 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), 4.2.3600.1 x2, 4.0.2880.1 x2, \(\Q(i, \sqrt{5})\), 8.0.5971968000.1 x2, 8.0.5971968000.2 x2, 8.0.207360000.5

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 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$