↑

Properties

Label 16.0.19268364558...9821.2
Degree $16$
Signature $[0, 8]$
Discriminant $19^{10}\cdot 61^{7}$
Root discriminant $38.05$
Ramified primes $19, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![311, 1117, 1504, 1654, 2625, 332, 964, -746, -63, -80, 55, 32, 12, -35, 13, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 13*x^14 - 35*x^13 + 12*x^12 + 32*x^11 + 55*x^10 - 80*x^9 - 63*x^8 - 746*x^7 + 964*x^6 + 332*x^5 + 2625*x^4 + 1654*x^3 + 1504*x^2 + 1117*x + 311)
 
gp: K = bnfinit(x^16 - 3*x^15 + 13*x^14 - 35*x^13 + 12*x^12 + 32*x^11 + 55*x^10 - 80*x^9 - 63*x^8 - 746*x^7 + 964*x^6 + 332*x^5 + 2625*x^4 + 1654*x^3 + 1504*x^2 + 1117*x + 311, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 13 x^{14} - 35 x^{13} + 12 x^{12} + 32 x^{11} + 55 x^{10} - 80 x^{9} - 63 x^{8} - 746 x^{7} + 964 x^{6} + 332 x^{5} + 2625 x^{4} + 1654 x^{3} + 1504 x^{2} + 1117 x + 311 \)

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:  \(19268364558874174255049821=19^{10}\cdot 61^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.05$
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:  $19, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{1384989473310962393032823} a^{15} + \frac{7036239947209951953930}{197855639044423199004689} a^{14} - \frac{93478070675053320101323}{1384989473310962393032823} a^{13} + \frac{21395237695980689224895}{197855639044423199004689} a^{12} + \frac{25323635644839663840964}{1384989473310962393032823} a^{11} + \frac{500570181510465455699850}{1384989473310962393032823} a^{10} + \frac{350439069179079330263802}{1384989473310962393032823} a^{9} + \frac{437653227305967533281373}{1384989473310962393032823} a^{8} - \frac{657201174439905657214801}{1384989473310962393032823} a^{7} + \frac{31636734775688844484187}{81469969018291905472519} a^{6} - \frac{21286723138902976422102}{1384989473310962393032823} a^{5} + \frac{592281800664398743364954}{1384989473310962393032823} a^{4} + \frac{202828734433684644579683}{1384989473310962393032823} a^{3} - \frac{540675956618052261941096}{1384989473310962393032823} a^{2} + \frac{26888874150933717845248}{197855639044423199004689} a - \frac{601068519868023779690665}{1384989473310962393032823}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 915113.543204 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.22021.1, 8.0.29580390901.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $16$ $16$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ $16$

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
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$