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Properties

Label 16.4.75192323506...1344.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{62}\cdot 113^{4}$
Root discriminant $47.84$
Ramified primes $2, 113$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-862, 800, 8736, 1008, -13748, -5760, 8088, 5104, -1043, -2056, -28, 184, -42, -8, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 8*x^13 - 42*x^12 + 184*x^11 - 28*x^10 - 2056*x^9 - 1043*x^8 + 5104*x^7 + 8088*x^6 - 5760*x^5 - 13748*x^4 + 1008*x^3 + 8736*x^2 + 800*x - 862)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 8*x^13 - 42*x^12 + 184*x^11 - 28*x^10 - 2056*x^9 - 1043*x^8 + 5104*x^7 + 8088*x^6 - 5760*x^5 - 13748*x^4 + 1008*x^3 + 8736*x^2 + 800*x - 862, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 8 x^{13} - 42 x^{12} + 184 x^{11} - 28 x^{10} - 2056 x^{9} - 1043 x^{8} + 5104 x^{7} + 8088 x^{6} - 5760 x^{5} - 13748 x^{4} + 1008 x^{3} + 8736 x^{2} + 800 x - 862 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(751923235065182967870521344=2^{62}\cdot 113^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.84$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{119} a^{14} - \frac{30}{119} a^{13} - \frac{3}{119} a^{12} - \frac{39}{119} a^{11} + \frac{9}{119} a^{10} - \frac{33}{119} a^{9} + \frac{25}{119} a^{8} - \frac{59}{119} a^{7} - \frac{41}{119} a^{6} + \frac{12}{119} a^{5} + \frac{8}{119} a^{4} + \frac{29}{119} a^{3} + \frac{23}{119} a^{2} - \frac{27}{119} a + \frac{47}{119}$, $\frac{1}{567929520214568031530730501719} a^{15} - \frac{1765630391010251017286004118}{567929520214568031530730501719} a^{14} + \frac{130029339789111273634572541248}{567929520214568031530730501719} a^{13} + \frac{9565003936299465757199850233}{81132788602081147361532928817} a^{12} + \frac{86360409194681473586158993917}{567929520214568031530730501719} a^{11} - \frac{70361115795507011044829264}{998118664700471057171758351} a^{10} - \frac{42002064520414946026796319863}{567929520214568031530730501719} a^{9} + \frac{93399504463040055071651073656}{567929520214568031530730501719} a^{8} + \frac{204494116151299286853763338807}{567929520214568031530730501719} a^{7} - \frac{202425682038719962348333803416}{567929520214568031530730501719} a^{6} + \frac{159440558074823349512465864523}{567929520214568031530730501719} a^{5} - \frac{58002702351068460060205863311}{567929520214568031530730501719} a^{4} - \frac{30293592837147273453178462702}{567929520214568031530730501719} a^{3} - \frac{19043805801880633744796597454}{567929520214568031530730501719} a^{2} + \frac{277633821698832453841496249305}{567929520214568031530730501719} a + \frac{214086994281357171467687118650}{567929520214568031530730501719}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 28665258.3528 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.242665652224.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.17$x^{8} + 16 x^{5} + 12 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8:C_2$$[2, 3, 4, 5]$
2.8.31.17$x^{8} + 16 x^{5} + 12 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8:C_2$$[2, 3, 4, 5]$
113Data not computed