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Properties

Label 16.4.26968313608...625.29
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 101^{10}$
Root discriminant $59.83$
Ramified primes $5, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53605, -41745, 64840, -10360, 10896, -3005, 18943, -493, -10296, 6427, -384, -1516, 397, 134, -39, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 39*x^14 + 134*x^13 + 397*x^12 - 1516*x^11 - 384*x^10 + 6427*x^9 - 10296*x^8 - 493*x^7 + 18943*x^6 - 3005*x^5 + 10896*x^4 - 10360*x^3 + 64840*x^2 - 41745*x + 53605)
 
gp: K = bnfinit(x^16 - 4*x^15 - 39*x^14 + 134*x^13 + 397*x^12 - 1516*x^11 - 384*x^10 + 6427*x^9 - 10296*x^8 - 493*x^7 + 18943*x^6 - 3005*x^5 + 10896*x^4 - 10360*x^3 + 64840*x^2 - 41745*x + 53605, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 39 x^{14} + 134 x^{13} + 397 x^{12} - 1516 x^{11} - 384 x^{10} + 6427 x^{9} - 10296 x^{8} - 493 x^{7} + 18943 x^{6} - 3005 x^{5} + 10896 x^{4} - 10360 x^{3} + 64840 x^{2} - 41745 x + 53605 \)

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:  \(26968313608671985107666015625=5^{12}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.83$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} - \frac{1}{5} a^{11} + \frac{2}{15} a^{10} + \frac{2}{5} a^{8} - \frac{7}{15} a^{6} + \frac{2}{5} a^{5} + \frac{1}{15} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{1185} a^{13} - \frac{8}{1185} a^{12} - \frac{238}{1185} a^{11} + \frac{58}{237} a^{10} + \frac{132}{395} a^{9} - \frac{25}{79} a^{8} + \frac{188}{1185} a^{7} + \frac{386}{1185} a^{6} + \frac{556}{1185} a^{5} - \frac{2}{237} a^{4} + \frac{59}{237} a^{3} + \frac{61}{237} a^{2} - \frac{70}{237} a + \frac{25}{237}$, $\frac{1}{712185} a^{14} - \frac{28}{142437} a^{13} + \frac{7488}{237395} a^{12} - \frac{136327}{712185} a^{11} + \frac{213968}{712185} a^{10} + \frac{16421}{237395} a^{9} + \frac{303989}{712185} a^{8} - \frac{7730}{142437} a^{7} + \frac{5769}{237395} a^{6} + \frac{137054}{712185} a^{5} - \frac{142639}{712185} a^{4} + \frac{15647}{47479} a^{3} + \frac{12971}{142437} a^{2} + \frac{42287}{142437} a - \frac{51569}{142437}$, $\frac{1}{3014581760905415438555343870675} a^{15} - \frac{78836789129779185650999}{200972117393694362570356258045} a^{14} + \frac{499756896749633314012663561}{3014581760905415438555343870675} a^{13} + \frac{92683117534968894753940667923}{3014581760905415438555343870675} a^{12} - \frac{646258790408334597080784274556}{3014581760905415438555343870675} a^{11} - \frac{296123242816230790531604180609}{602916352181083087711068774135} a^{10} - \frac{406199815369325353710518415739}{3014581760905415438555343870675} a^{9} + \frac{466026684621685103912843881912}{1004860586968471812851781290225} a^{8} + \frac{554549846617268986159751709038}{3014581760905415438555343870675} a^{7} + \frac{290371722522048161762441202973}{1004860586968471812851781290225} a^{6} + \frac{892374906274382455867312936559}{3014581760905415438555343870675} a^{5} + \frac{997979849053760940637608331046}{3014581760905415438555343870675} a^{4} - \frac{10953808389419565360166824086}{40194423478738872514071251609} a^{3} + \frac{283673889582974018160906888533}{602916352181083087711068774135} a^{2} + \frac{11781335681152473500160292262}{40194423478738872514071251609} a - \frac{161233168123702552837158309224}{602916352181083087711068774135}$

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:  \( 99884309.1227 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.4.1625943765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
101Data not computed