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Properties

Label 16.8.58454733517...8864.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{40}\cdot 3^{4}\cdot 17^{8}\cdot 97^{2}$
Root discriminant $54.38$
Ramified primes $2, 3, 17, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T608)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3742, 29192, -67704, 41456, 26290, -46244, 14544, 6192, -517, 620, -1816, -8, -2, -8, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 20*x^14 - 8*x^13 - 2*x^12 - 8*x^11 - 1816*x^10 + 620*x^9 - 517*x^8 + 6192*x^7 + 14544*x^6 - 46244*x^5 + 26290*x^4 + 41456*x^3 - 67704*x^2 + 29192*x - 3742)
 
gp: K = bnfinit(x^16 + 20*x^14 - 8*x^13 - 2*x^12 - 8*x^11 - 1816*x^10 + 620*x^9 - 517*x^8 + 6192*x^7 + 14544*x^6 - 46244*x^5 + 26290*x^4 + 41456*x^3 - 67704*x^2 + 29192*x - 3742, 1)
 

Normalized defining polynomial

\( x^{16} + 20 x^{14} - 8 x^{13} - 2 x^{12} - 8 x^{11} - 1816 x^{10} + 620 x^{9} - 517 x^{8} + 6192 x^{7} + 14544 x^{6} - 46244 x^{5} + 26290 x^{4} + 41456 x^{3} - 67704 x^{2} + 29192 x - 3742 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5845473351728475416888868864=2^{40}\cdot 3^{4}\cdot 17^{8}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.38$
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:  $2, 3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{7} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{21} a^{12} + \frac{4}{21} a^{9} + \frac{2}{7} a^{8} - \frac{10}{21} a^{7} + \frac{8}{21} a^{6} + \frac{3}{7} a^{5} - \frac{2}{21} a^{4} - \frac{5}{21} a^{3} + \frac{8}{21} a^{2} - \frac{1}{21} a - \frac{4}{21}$, $\frac{1}{21} a^{13} + \frac{1}{21} a^{10} - \frac{1}{7} a^{9} - \frac{10}{21} a^{8} - \frac{1}{3} a^{7} + \frac{3}{7} a^{6} - \frac{5}{21} a^{5} + \frac{4}{21} a^{4} - \frac{1}{3} a^{3} + \frac{5}{21} a^{2} - \frac{4}{21} a + \frac{2}{7}$, $\frac{1}{160923} a^{14} - \frac{1030}{53641} a^{13} - \frac{2258}{160923} a^{12} - \frac{878}{160923} a^{11} - \frac{2881}{53641} a^{10} + \frac{37}{553} a^{9} - \frac{328}{22989} a^{8} - \frac{4849}{160923} a^{7} - \frac{1628}{53641} a^{6} - \frac{46325}{160923} a^{5} - \frac{374}{7663} a^{4} - \frac{3566}{7663} a^{3} - \frac{7646}{22989} a^{2} - \frac{3631}{160923} a + \frac{3185}{22989}$, $\frac{1}{17306510449437614780196594297} a^{15} - \frac{3827249301206124182245}{2472358635633944968599513471} a^{14} + \frac{42040899936652036741417738}{17306510449437614780196594297} a^{13} - \frac{358803885481168010583321119}{17306510449437614780196594297} a^{12} + \frac{704702629044320608109029145}{17306510449437614780196594297} a^{11} + \frac{274601879401569999251703726}{5768836816479204926732198099} a^{10} - \frac{1749666477739766994364699370}{5768836816479204926732198099} a^{9} + \frac{1774609060580879927059497928}{5768836816479204926732198099} a^{8} - \frac{366077734120883192526418688}{5768836816479204926732198099} a^{7} - \frac{222658162018494325941544930}{824119545211314989533171157} a^{6} + \frac{7186523834462295865348081343}{17306510449437614780196594297} a^{5} - \frac{5975094916242126078786225575}{17306510449437614780196594297} a^{4} - \frac{6594459429069400475659311940}{17306510449437614780196594297} a^{3} + \frac{1272058741061147972519489886}{5768836816479204926732198099} a^{2} - \frac{14000008022962161036792166}{17306510449437614780196594297} a - \frac{99041581174503443052201088}{824119545211314989533171157}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $11$
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:  \( 56202701.5395 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T608):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.4.4352.1 x2, 4.4.9248.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.5473632256.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.20.53$x^{8} + 4 x^{5} + 2 x^{4} + 2$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
2.8.20.53$x^{8} + 4 x^{5} + 2 x^{4} + 2$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$