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Properties

Label 16.8.49023798342...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 439^{4}\cdot 2411^{4}$
Root discriminant $71.72$
Ramified primes $5, 439, 2411$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32715509, -90190604, 112248421, -77068202, 23736488, 3245050, -5668399, 1946502, -23894, -135524, 40346, -1659, -966, 307, -40, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 40*x^14 + 307*x^13 - 966*x^12 - 1659*x^11 + 40346*x^10 - 135524*x^9 - 23894*x^8 + 1946502*x^7 - 5668399*x^6 + 3245050*x^5 + 23736488*x^4 - 77068202*x^3 + 112248421*x^2 - 90190604*x + 32715509)
 
gp: K = bnfinit(x^16 - 5*x^15 - 40*x^14 + 307*x^13 - 966*x^12 - 1659*x^11 + 40346*x^10 - 135524*x^9 - 23894*x^8 + 1946502*x^7 - 5668399*x^6 + 3245050*x^5 + 23736488*x^4 - 77068202*x^3 + 112248421*x^2 - 90190604*x + 32715509, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 40 x^{14} + 307 x^{13} - 966 x^{12} - 1659 x^{11} + 40346 x^{10} - 135524 x^{9} - 23894 x^{8} + 1946502 x^{7} - 5668399 x^{6} + 3245050 x^{5} + 23736488 x^{4} - 77068202 x^{3} + 112248421 x^{2} - 90190604 x + 32715509 \)

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:  \(490237983424834767085031640625=5^{8}\cdot 439^{4}\cdot 2411^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.72$
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:  $5, 439, 2411$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{31} a^{13} - \frac{6}{31} a^{12} - \frac{11}{31} a^{11} - \frac{2}{31} a^{10} - \frac{1}{31} a^{9} - \frac{12}{31} a^{8} - \frac{4}{31} a^{7} + \frac{11}{31} a^{6} + \frac{11}{31} a^{5} - \frac{10}{31} a^{4} + \frac{10}{31} a^{3} - \frac{9}{31} a^{2} - \frac{13}{31} a$, $\frac{1}{31} a^{14} + \frac{15}{31} a^{12} - \frac{6}{31} a^{11} - \frac{13}{31} a^{10} + \frac{13}{31} a^{9} - \frac{14}{31} a^{8} - \frac{13}{31} a^{7} + \frac{15}{31} a^{6} - \frac{6}{31} a^{5} + \frac{12}{31} a^{4} - \frac{11}{31} a^{3} - \frac{5}{31} a^{2} + \frac{15}{31} a$, $\frac{1}{50080582386434037751890847282516038743153993027278986451} a^{15} - \frac{25290260966117037485566713865966527333922307279381081}{1615502657626904443609382170403743185263032033138031821} a^{14} - \frac{331321474558152862826153057489748338867503143456497901}{50080582386434037751890847282516038743153993027278986451} a^{13} - \frac{16506764680948901745570290008731959729647310766775026602}{50080582386434037751890847282516038743153993027278986451} a^{12} + \frac{24369705033135746438382505820140454878203521497216155377}{50080582386434037751890847282516038743153993027278986451} a^{11} - \frac{2831786131528564557190328590113783986905270162850109543}{50080582386434037751890847282516038743153993027278986451} a^{10} + \frac{11459929498680987389540438130686719734457967171565093980}{50080582386434037751890847282516038743153993027278986451} a^{9} - \frac{22685299595393795059416784139894636073092894894679207699}{50080582386434037751890847282516038743153993027278986451} a^{8} + \frac{354358902668797749859791672056155282051756285694263151}{50080582386434037751890847282516038743153993027278986451} a^{7} + \frac{871678871522953069683062570689708816269830930328238413}{50080582386434037751890847282516038743153993027278986451} a^{6} + \frac{8096602229299051846801061113214089257457989701653961597}{50080582386434037751890847282516038743153993027278986451} a^{5} + \frac{6747724260950942579201340983769575434485978618515348317}{50080582386434037751890847282516038743153993027278986451} a^{4} - \frac{4060069294186262068024787647645899326727085753164740755}{50080582386434037751890847282516038743153993027278986451} a^{3} - \frac{19646514298377128044462493453525102874951448570011438594}{50080582386434037751890847282516038743153993027278986451} a^{2} - \frac{17986479160254027934775389392664143106490458378019225406}{50080582386434037751890847282516038743153993027278986451} a + \frac{229183292823743638440301995724698850119551660791196217}{1615502657626904443609382170403743185263032033138031821}$

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:  $11$
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:  \( 578154887.404 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1869 are not computed
Character table for t16n1869 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.661518125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
439Data not computed
2411Data not computed