↑

Properties

Label 16.8.30775746840...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 13^{2}\cdot 29^{6}\cdot 97^{8}$
Root discriminant $107.28$
Ramified primes $5, 13, 29, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1220

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4669264, -8384456, 9752220, -1509366, -2534855, 4548148, -2436777, 628878, 41755, -130094, 22853, 2486, -1021, 252, -16, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 16*x^14 + 252*x^13 - 1021*x^12 + 2486*x^11 + 22853*x^10 - 130094*x^9 + 41755*x^8 + 628878*x^7 - 2436777*x^6 + 4548148*x^5 - 2534855*x^4 - 1509366*x^3 + 9752220*x^2 - 8384456*x + 4669264)
 
gp: K = bnfinit(x^16 - 8*x^15 - 16*x^14 + 252*x^13 - 1021*x^12 + 2486*x^11 + 22853*x^10 - 130094*x^9 + 41755*x^8 + 628878*x^7 - 2436777*x^6 + 4548148*x^5 - 2534855*x^4 - 1509366*x^3 + 9752220*x^2 - 8384456*x + 4669264, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 16 x^{14} + 252 x^{13} - 1021 x^{12} + 2486 x^{11} + 22853 x^{10} - 130094 x^{9} + 41755 x^{8} + 628878 x^{7} - 2436777 x^{6} + 4548148 x^{5} - 2534855 x^{4} - 1509366 x^{3} + 9752220 x^{2} - 8384456 x + 4669264 \)

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:  \(307757468400156944014204800390625=5^{8}\cdot 13^{2}\cdot 29^{6}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.28$
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:  $5, 13, 29, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{348} a^{12} - \frac{1}{58} a^{11} - \frac{5}{116} a^{10} + \frac{7}{174} a^{9} - \frac{11}{174} a^{8} - \frac{1}{87} a^{7} - \frac{5}{116} a^{6} + \frac{1}{87} a^{5} + \frac{6}{29} a^{4} - \frac{14}{29} a^{3} - \frac{55}{116} a^{2} - \frac{40}{87} a - \frac{17}{87}$, $\frac{1}{348} a^{13} + \frac{7}{348} a^{11} - \frac{3}{58} a^{10} + \frac{1}{87} a^{9} - \frac{5}{87} a^{8} + \frac{77}{348} a^{7} - \frac{43}{174} a^{6} - \frac{5}{87} a^{5} - \frac{13}{174} a^{4} - \frac{13}{348} a^{3} + \frac{21}{58} a^{2} - \frac{25}{87} a + \frac{43}{87}$, $\frac{1}{3121005791527395096} a^{14} - \frac{7}{3121005791527395096} a^{13} - \frac{1242452739113701}{1040335263842465032} a^{12} + \frac{22364149304046709}{3121005791527395096} a^{11} - \frac{8457912190049349}{260083815960616258} a^{10} - \frac{8372984710462817}{120038684289515196} a^{9} - \frac{196633578742117073}{3121005791527395096} a^{8} + \frac{86018348153417255}{1040335263842465032} a^{7} + \frac{102143394625127363}{780251447881848774} a^{6} + \frac{162687602985773897}{520167631921232516} a^{5} + \frac{1327854009736531405}{3121005791527395096} a^{4} + \frac{188276680017310479}{1040335263842465032} a^{3} - \frac{52780962614067881}{120038684289515196} a^{2} - \frac{78201082805950075}{390125723940924387} a - \frac{135279807217526113}{390125723940924387}$, $\frac{1}{3467437434386935951656} a^{15} + \frac{137}{866859358596733987914} a^{14} + \frac{98341868731317073}{288953119532244662638} a^{13} + \frac{37114570458323075}{577906239064489325276} a^{12} - \frac{5512583477796733651}{315221584944266904696} a^{11} - \frac{366901394308637021}{26268465412022242058} a^{10} + \frac{8586051108512060383}{1155812478128978650552} a^{9} + \frac{105129114716854171309}{1733718717193467975828} a^{8} - \frac{689787043973237265229}{3467437434386935951656} a^{7} - \frac{2225685499233964275}{288953119532244662638} a^{6} + \frac{955014462804192089879}{3467437434386935951656} a^{5} - \frac{45790714992213755569}{433429679298366993957} a^{4} + \frac{73656350002205663053}{3467437434386935951656} a^{3} - \frac{368978980442982804395}{866859358596733987914} a^{2} + \frac{257730115378885639117}{866859358596733987914} a + \frac{269317683531406630}{14945851010288517033}$

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:  \( 27609215854.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{485}) \), \(\Q(\sqrt{97}) \), 4.4.6821525.1, 4.4.725.1, \(\Q(\sqrt{5}, \sqrt{97})\), 8.8.46533203325625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$