↑

Properties

Label 16.0.16226472272...9456.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 7^{8}$
Root discriminant $21.17$
Ramified primes $2, 7$
Class number $4$
Class group $[2, 2]$
Galois group $(C_2\times D_8):C_2$ (as 16T126)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 52, 40, 212, 160, 312, 152, 272, -8, 76, -48, 32, -16, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 - 16*x^13 + 32*x^12 - 48*x^11 + 76*x^10 - 8*x^9 + 272*x^8 + 152*x^7 + 312*x^6 + 160*x^5 + 212*x^4 + 40*x^3 + 52*x^2 + 1)
 
gp: K = bnfinit(x^16 + 8*x^14 - 16*x^13 + 32*x^12 - 48*x^11 + 76*x^10 - 8*x^9 + 272*x^8 + 152*x^7 + 312*x^6 + 160*x^5 + 212*x^4 + 40*x^3 + 52*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} - 16 x^{13} + 32 x^{12} - 48 x^{11} + 76 x^{10} - 8 x^{9} + 272 x^{8} + 152 x^{7} + 312 x^{6} + 160 x^{5} + 212 x^{4} + 40 x^{3} + 52 x^{2} + 1 \)

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:  \(1622647227216566419456=2^{48}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.17$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{203} a^{13} - \frac{6}{203} a^{12} + \frac{90}{203} a^{11} + \frac{1}{203} a^{10} + \frac{67}{203} a^{9} + \frac{36}{203} a^{8} + \frac{5}{203} a^{7} - \frac{7}{29} a^{6} + \frac{44}{203} a^{5} - \frac{4}{29} a^{4} + \frac{83}{203} a^{3} - \frac{64}{203} a^{2} - \frac{31}{203} a + \frac{40}{203}$, $\frac{1}{429751} a^{14} - \frac{149}{429751} a^{13} - \frac{10884}{429751} a^{12} + \frac{23955}{61393} a^{11} + \frac{116185}{429751} a^{10} + \frac{114778}{429751} a^{9} - \frac{146779}{429751} a^{8} + \frac{112104}{429751} a^{7} - \frac{61041}{429751} a^{6} + \frac{130415}{429751} a^{5} - \frac{15401}{429751} a^{4} - \frac{117899}{429751} a^{3} - \frac{72021}{429751} a^{2} - \frac{151721}{429751} a + \frac{128550}{429751}$, $\frac{1}{1483070701} a^{15} + \frac{1042}{1483070701} a^{14} - \frac{1416203}{1483070701} a^{13} - \frac{43553052}{1483070701} a^{12} + \frac{69798646}{1483070701} a^{11} - \frac{12702321}{30266749} a^{10} - \frac{2328394}{87239453} a^{9} - \frac{94033279}{211867243} a^{8} + \frac{430719729}{1483070701} a^{7} + \frac{105961428}{1483070701} a^{6} + \frac{419482943}{1483070701} a^{5} - \frac{410948}{1483070701} a^{4} - \frac{455128855}{1483070701} a^{3} + \frac{611372962}{1483070701} a^{2} - \frac{350322806}{1483070701} a - \frac{688656373}{1483070701}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( -1 \) (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:  \( 8142.79016471 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_8):C_2$ (as 16T126):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $(C_2\times D_8):C_2$
Character table for $(C_2\times D_8):C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.1792.1, 4.2.50176.1, 4.0.7168.1, 8.2.1438646272.4, 8.2.359661568.2, 8.0.10070523904.8

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$