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Properties

Label 16.16.1604279675...5625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{8}\cdot 29^{6}\cdot 1621^{4}$
Root discriminant $50.16$
Ramified primes $5, 29, 1621$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1276

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-12661, 74332, -26268, -288997, 87056, 291115, -76921, -126127, 32002, 27489, -6921, -3142, 791, 179, -45, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 45*x^14 + 179*x^13 + 791*x^12 - 3142*x^11 - 6921*x^10 + 27489*x^9 + 32002*x^8 - 126127*x^7 - 76921*x^6 + 291115*x^5 + 87056*x^4 - 288997*x^3 - 26268*x^2 + 74332*x - 12661)
 
gp: K = bnfinit(x^16 - 4*x^15 - 45*x^14 + 179*x^13 + 791*x^12 - 3142*x^11 - 6921*x^10 + 27489*x^9 + 32002*x^8 - 126127*x^7 - 76921*x^6 + 291115*x^5 + 87056*x^4 - 288997*x^3 - 26268*x^2 + 74332*x - 12661, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 45 x^{14} + 179 x^{13} + 791 x^{12} - 3142 x^{11} - 6921 x^{10} + 27489 x^{9} + 32002 x^{8} - 126127 x^{7} - 76921 x^{6} + 291115 x^{5} + 87056 x^{4} - 288997 x^{3} - 26268 x^{2} + 74332 x - 12661 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1604279675444921972578515625=5^{8}\cdot 29^{6}\cdot 1621^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.16$
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:  $5, 29, 1621$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} - \frac{7}{19} a^{13} - \frac{8}{19} a^{12} - \frac{4}{19} a^{11} - \frac{9}{19} a^{10} - \frac{6}{19} a^{9} + \frac{2}{19} a^{8} + \frac{8}{19} a^{7} - \frac{5}{19} a^{6} + \frac{5}{19} a^{5} - \frac{9}{19} a^{4} + \frac{9}{19} a^{3} - \frac{2}{19} a^{2} - \frac{9}{19} a + \frac{4}{19}$, $\frac{1}{8427776650230906556336501201} a^{15} - \frac{194113216278311048083760823}{8427776650230906556336501201} a^{14} - \frac{85425506679839386906816218}{8427776650230906556336501201} a^{13} - \frac{3175722863467376180150405609}{8427776650230906556336501201} a^{12} + \frac{31660573888660308249709605}{142843672037811975531127139} a^{11} - \frac{1307280649009844764865962139}{8427776650230906556336501201} a^{10} + \frac{1383691298089616582547652000}{8427776650230906556336501201} a^{9} - \frac{2386588783726987939455268045}{8427776650230906556336501201} a^{8} - \frac{2206340478684081451931515040}{8427776650230906556336501201} a^{7} + \frac{498563806893471245507120300}{8427776650230906556336501201} a^{6} + \frac{100757953373108028077400152}{8427776650230906556336501201} a^{5} + \frac{2858543152002584126652297302}{8427776650230906556336501201} a^{4} + \frac{659357227243857599782423382}{8427776650230906556336501201} a^{3} - \frac{4179888299408688440977943909}{8427776650230906556336501201} a^{2} - \frac{3911872358275686865547637181}{8427776650230906556336501201} a - \frac{3200879091599908166094445}{7322134361625461821317551}$

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:  $15$
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:  \( 213699668.471 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1276:

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 t16n1276
Character table for t16n1276 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.852038125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
1621Data not computed