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Properties

Label 16.0.24956872719...3125.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{11}\cdot 59^{10}$
Root discriminant $38.67$
Ramified primes $5, 59$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8275, -5850, 2730, -19510, 19309, -10587, 14235, 2795, 3056, -3929, 14, 81, 120, 11, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 8*x^14 + 11*x^13 + 120*x^12 + 81*x^11 + 14*x^10 - 3929*x^9 + 3056*x^8 + 2795*x^7 + 14235*x^6 - 10587*x^5 + 19309*x^4 - 19510*x^3 + 2730*x^2 - 5850*x + 8275)
 
gp: K = bnfinit(x^16 - 4*x^15 - 8*x^14 + 11*x^13 + 120*x^12 + 81*x^11 + 14*x^10 - 3929*x^9 + 3056*x^8 + 2795*x^7 + 14235*x^6 - 10587*x^5 + 19309*x^4 - 19510*x^3 + 2730*x^2 - 5850*x + 8275, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 8 x^{14} + 11 x^{13} + 120 x^{12} + 81 x^{11} + 14 x^{10} - 3929 x^{9} + 3056 x^{8} + 2795 x^{7} + 14235 x^{6} - 10587 x^{5} + 19309 x^{4} - 19510 x^{3} + 2730 x^{2} - 5850 x + 8275 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24956872719757880908203125=5^{11}\cdot 59^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.67$
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:  $5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{32655} a^{14} - \frac{47}{4665} a^{13} - \frac{1049}{32655} a^{12} + \frac{29}{32655} a^{11} - \frac{73}{4665} a^{10} + \frac{2249}{10885} a^{9} + \frac{4924}{32655} a^{8} + \frac{1314}{10885} a^{7} + \frac{132}{311} a^{6} - \frac{4672}{32655} a^{5} - \frac{15128}{32655} a^{4} + \frac{16192}{32655} a^{3} - \frac{13778}{32655} a^{2} - \frac{68}{933} a - \frac{950}{6531}$, $\frac{1}{738453519354600761342208707955} a^{15} + \frac{426560745016938456487759}{35164453302600036254390890855} a^{14} + \frac{4219909572606162057207452187}{246151173118200253780736235985} a^{13} - \frac{32280748672781036981576604902}{738453519354600761342208707955} a^{12} + \frac{581213051875531391748010099}{7032890660520007250878178171} a^{11} - \frac{8985264561864399711438543493}{147690703870920152268441741591} a^{10} + \frac{220156583314255224176241446902}{738453519354600761342208707955} a^{9} + \frac{35301114013360467430147277366}{738453519354600761342208707955} a^{8} - \frac{14189746916363350680158509366}{35164453302600036254390890855} a^{7} - \frac{187689345211467354395347587796}{738453519354600761342208707955} a^{6} + \frac{22116372996914001772525055683}{147690703870920152268441741591} a^{5} - \frac{100430022597759078813018073436}{246151173118200253780736235985} a^{4} + \frac{116055431439976458356202686407}{246151173118200253780736235985} a^{3} + \frac{9810949676141959288514293703}{21098671981560021752634534513} a^{2} + \frac{1698956586313577652180420661}{49230234623640050756147247197} a + \frac{6246651531842050460416349}{63742211424652633693759923}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{6}$, which has order $6$ (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:  $7$
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:  \( 248007.028891 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-59}) \), 4.0.17405.1, 8.0.1514670125.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ 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
$5$5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$59$59.8.6.2$x^{8} + 177 x^{4} + 13924$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
59.8.4.1$x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$