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Properties

Label 16.0.23126068311...5856.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 11^{10}$
Root discriminant $28.86$
Ramified primes $2, 3, 11$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T329)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![484, 484, -550, -616, -160, -130, 555, -624, 1078, -426, 225, 38, -49, 20, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 4*x^14 + 20*x^13 - 49*x^12 + 38*x^11 + 225*x^10 - 426*x^9 + 1078*x^8 - 624*x^7 + 555*x^6 - 130*x^5 - 160*x^4 - 616*x^3 - 550*x^2 + 484*x + 484)
 
gp: K = bnfinit(x^16 - 2*x^15 - 4*x^14 + 20*x^13 - 49*x^12 + 38*x^11 + 225*x^10 - 426*x^9 + 1078*x^8 - 624*x^7 + 555*x^6 - 130*x^5 - 160*x^4 - 616*x^3 - 550*x^2 + 484*x + 484, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 4 x^{14} + 20 x^{13} - 49 x^{12} + 38 x^{11} + 225 x^{10} - 426 x^{9} + 1078 x^{8} - 624 x^{7} + 555 x^{6} - 130 x^{5} - 160 x^{4} - 616 x^{3} - 550 x^{2} + 484 x + 484 \)

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:  \(231260683111582301945856=2^{24}\cdot 3^{12}\cdot 11^{10}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{18} a^{12} + \frac{1}{9} a^{11} + \frac{1}{18} a^{10} - \frac{1}{9} a^{9} - \frac{4}{9} a^{8} - \frac{2}{9} a^{7} + \frac{5}{18} a^{6} - \frac{2}{9} a^{5} - \frac{5}{18} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{54} a^{13} + \frac{1}{54} a^{12} - \frac{1}{54} a^{11} - \frac{1}{18} a^{10} - \frac{1}{9} a^{9} + \frac{11}{27} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{19}{54} a^{5} + \frac{7}{18} a^{4} - \frac{2}{9} a^{3} - \frac{10}{27} a^{2} + \frac{13}{27} a - \frac{5}{27}$, $\frac{1}{594} a^{14} - \frac{1}{297} a^{13} - \frac{2}{297} a^{12} - \frac{5}{33} a^{11} + \frac{13}{198} a^{10} + \frac{74}{297} a^{9} - \frac{79}{198} a^{8} + \frac{13}{33} a^{7} + \frac{11}{27} a^{6} - \frac{38}{99} a^{5} - \frac{13}{198} a^{4} + \frac{89}{297} a^{3} - \frac{146}{297} a^{2} + \frac{5}{27} a + \frac{4}{9}$, $\frac{1}{6711108307097753394} a^{15} + \frac{4796168388575645}{6711108307097753394} a^{14} - \frac{7843304352852677}{6711108307097753394} a^{13} + \frac{22171568706356666}{1118518051182958899} a^{12} - \frac{424894700739561407}{3355554153548876697} a^{11} - \frac{191808569779666601}{6711108307097753394} a^{10} + \frac{1067472512255542219}{6711108307097753394} a^{9} - \frac{277297509075612061}{610100755190704854} a^{8} - \frac{2705564369917840897}{6711108307097753394} a^{7} + \frac{388860295244623462}{3355554153548876697} a^{6} - \frac{1243357534603545146}{3355554153548876697} a^{5} - \frac{641439376855245695}{6711108307097753394} a^{4} - \frac{15648340366369072}{1118518051182958899} a^{3} - \frac{838558197126779048}{3355554153548876697} a^{2} - \frac{120571552788722939}{305050377595352427} a + \frac{66924086035017326}{305050377595352427}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 530457.830501 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T329):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{33}) \), 4.0.13068.1 x2, \(\Q(\sqrt{3}, \sqrt{11})\), 4.0.4752.1 x2, 8.0.2732361984.4

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$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$$[\ ]_{4}^{2}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$