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Properties

Label 16.0.89161004482...000.15
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 5^{8}$
Root discriminant $20.39$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $C_2 \times (C_4\times C_2):C_2$ (as 16T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 0, -3000, 0, 5275, 0, -4320, 0, 2109, 0, -624, 0, 115, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 115*x^12 - 624*x^10 + 2109*x^8 - 4320*x^6 + 5275*x^4 - 3000*x^2 + 625)
 
gp: K = bnfinit(x^16 - 12*x^14 + 115*x^12 - 624*x^10 + 2109*x^8 - 4320*x^6 + 5275*x^4 - 3000*x^2 + 625, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{14} + 115 x^{12} - 624 x^{10} + 2109 x^{8} - 4320 x^{6} + 5275 x^{4} - 3000 x^{2} + 625 \)

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:  \(891610044825600000000=2^{32}\cdot 3^{12}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.39$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{9} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{12} - \frac{2}{25} a^{10} - \frac{1}{5} a^{8} + \frac{1}{25} a^{6} - \frac{6}{25} a^{4} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{13} - \frac{2}{25} a^{11} - \frac{1}{5} a^{9} + \frac{1}{25} a^{7} - \frac{6}{25} a^{5} - \frac{1}{5} a^{3}$, $\frac{1}{2550065375} a^{14} + \frac{8181773}{2550065375} a^{12} + \frac{3868598}{39231775} a^{10} - \frac{88740948}{196158875} a^{8} - \frac{42367021}{231824125} a^{6} - \frac{12655066}{510013075} a^{4} + \frac{38654581}{102002615} a^{2} - \frac{4582596}{20400523}$, $\frac{1}{2550065375} a^{15} + \frac{8181773}{2550065375} a^{13} + \frac{3868598}{39231775} a^{11} - \frac{88740948}{196158875} a^{9} - \frac{42367021}{231824125} a^{7} - \frac{12655066}{510013075} a^{5} + \frac{38654581}{102002615} a^{3} - \frac{4582596}{20400523} a$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -\frac{9328001}{2550065375} a^{14} + \frac{95851222}{2550065375} a^{12} - \frac{14298754}{39231775} a^{10} + \frac{342650098}{196158875} a^{8} - \frac{1283169559}{231824125} a^{6} + \frac{5206926742}{510013075} a^{4} - \frac{1150864834}{102002615} a^{2} + \frac{60998362}{20400523} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 14466.6129733 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:C_2$ (as 16T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_4\times C_2):C_2$
Character table for $C_2 \times (C_4\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(i, \sqrt{30})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), 8.0.3317760000.2, 8.0.4665600.1, 8.0.1194393600.9

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$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}$