↑

Properties

Label 16.10.3193845368...4779.3
Degree $16$
Signature $[10, 3]$
Discriminant $-\,41^{15}\cdot 59^{3}$
Root discriminant $69.83$
Ramified primes $41, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![138713, 92599, -1189053, 930707, 198769, -268128, -75863, 55181, 27980, -5086, -4189, -526, 387, 116, -28, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 28*x^14 + 116*x^13 + 387*x^12 - 526*x^11 - 4189*x^10 - 5086*x^9 + 27980*x^8 + 55181*x^7 - 75863*x^6 - 268128*x^5 + 198769*x^4 + 930707*x^3 - 1189053*x^2 + 92599*x + 138713)
 
gp: K = bnfinit(x^16 - 5*x^15 - 28*x^14 + 116*x^13 + 387*x^12 - 526*x^11 - 4189*x^10 - 5086*x^9 + 27980*x^8 + 55181*x^7 - 75863*x^6 - 268128*x^5 + 198769*x^4 + 930707*x^3 - 1189053*x^2 + 92599*x + 138713, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 28 x^{14} + 116 x^{13} + 387 x^{12} - 526 x^{11} - 4189 x^{10} - 5086 x^{9} + 27980 x^{8} + 55181 x^{7} - 75863 x^{6} - 268128 x^{5} + 198769 x^{4} + 930707 x^{3} - 1189053 x^{2} + 92599 x + 138713 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-319384536834647064600389984779=-\,41^{15}\cdot 59^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.83$
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:  $41, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} + \frac{2}{5} a^{4} + \frac{1}{10} a^{3} + \frac{1}{5} a^{2} - \frac{3}{10} a + \frac{3}{10}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{2} a^{6} + \frac{2}{5} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{3}{10} a^{2} + \frac{3}{10} a$, $\frac{1}{230} a^{14} + \frac{11}{230} a^{13} + \frac{2}{115} a^{12} + \frac{9}{46} a^{11} - \frac{21}{230} a^{10} - \frac{1}{46} a^{9} + \frac{56}{115} a^{8} + \frac{24}{115} a^{7} + \frac{54}{115} a^{6} + \frac{11}{23} a^{5} - \frac{93}{230} a^{4} + \frac{2}{23} a^{3} + \frac{7}{230} a^{2} + \frac{5}{23} a - \frac{2}{5}$, $\frac{1}{211175624785996029247064219461320790} a^{15} - \frac{3258438903589059384206386872037}{211175624785996029247064219461320790} a^{14} + \frac{3009034556871790494773021848003753}{105587812392998014623532109730660395} a^{13} + \frac{4761218812927174919664319262966786}{105587812392998014623532109730660395} a^{12} - \frac{40069340721634186755168906800493161}{211175624785996029247064219461320790} a^{11} - \frac{5488702368937351356533584709521739}{42235124957199205849412843892264158} a^{10} + \frac{2871748490320876651507393382516416}{21117562478599602924706421946132079} a^{9} - \frac{21213594952032428286625348492961337}{211175624785996029247064219461320790} a^{8} - \frac{13496407270586941058355429585834887}{42235124957199205849412843892264158} a^{7} - \frac{1572753848808004403217616669264409}{5707449318540433222893627553008670} a^{6} + \frac{4637330976584420518469975514600016}{105587812392998014623532109730660395} a^{5} + \frac{7175597403589587766105310720034585}{21117562478599602924706421946132079} a^{4} - \frac{28496120030865537273689785045788257}{105587812392998014623532109730660395} a^{3} - \frac{9277813754174292842339195412779159}{105587812392998014623532109730660395} a^{2} - \frac{59151554629066527650597893987622459}{211175624785996029247064219461320790} a + \frac{31965255504561718476496357907203}{248149970371323183604070763174290}$

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:  $12$
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:  \( 2893408318.83 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.6.11490502158979.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ $16$ R

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
41Data not computed
59Data not computed