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Properties

Label 16.0.99827490879...8125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{9}\cdot 59^{10}$
Root discriminant $31.62$
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![35, -425, 1625, -1725, 1071, 226, -456, 310, 184, -244, 115, 49, -56, 18, 5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 5*x^14 + 18*x^13 - 56*x^12 + 49*x^11 + 115*x^10 - 244*x^9 + 184*x^8 + 310*x^7 - 456*x^6 + 226*x^5 + 1071*x^4 - 1725*x^3 + 1625*x^2 - 425*x + 35)
 
gp: K = bnfinit(x^16 - 4*x^15 + 5*x^14 + 18*x^13 - 56*x^12 + 49*x^11 + 115*x^10 - 244*x^9 + 184*x^8 + 310*x^7 - 456*x^6 + 226*x^5 + 1071*x^4 - 1725*x^3 + 1625*x^2 - 425*x + 35, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 5 x^{14} + 18 x^{13} - 56 x^{12} + 49 x^{11} + 115 x^{10} - 244 x^{9} + 184 x^{8} + 310 x^{7} - 456 x^{6} + 226 x^{5} + 1071 x^{4} - 1725 x^{3} + 1625 x^{2} - 425 x + 35 \)

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:  \(998274908790315236328125=5^{9}\cdot 59^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.62$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{4}{9} a^{8} + \frac{1}{3} a^{7} - \frac{4}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{2}{9} a^{7} - \frac{1}{9} a^{6} - \frac{2}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{5067428129254953} a^{15} + \frac{18755190830759}{1689142709751651} a^{14} + \frac{258606374071601}{5067428129254953} a^{13} - \frac{761067263360266}{5067428129254953} a^{12} + \frac{56850775011977}{563047569917217} a^{11} + \frac{311749011544144}{5067428129254953} a^{10} + \frac{767294115494366}{5067428129254953} a^{9} + \frac{570056515627744}{5067428129254953} a^{8} + \frac{58235754898292}{5067428129254953} a^{7} - \frac{38471182911103}{563047569917217} a^{6} + \frac{127849560218650}{1689142709751651} a^{5} + \frac{427163954902600}{5067428129254953} a^{4} - \frac{1386594101080949}{5067428129254953} a^{3} - \frac{68980675690934}{723918304179279} a^{2} - \frac{470510627708753}{1689142709751651} a + \frac{230051712080803}{723918304179279}$

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:  \( 81553.4663298 \) (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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ 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$5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$59$59.8.4.1$x^{8} + 97468 x^{4} - 205379 x^{2} + 2375002756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
59.8.6.2$x^{8} + 177 x^{4} + 13924$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$