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Properties

Label 16.8.15413672017...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 59^{8}$
Root discriminant $88.97$
Ramified primes $2, 3, 5, 59$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $(C_2\times D_4).C_2^3$ (as 16T315)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![72361, 0, -4890, 0, -80172, 0, -19600, 0, 9364, 0, 2010, 0, -113, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 - 113*x^12 + 2010*x^10 + 9364*x^8 - 19600*x^6 - 80172*x^4 - 4890*x^2 + 72361)
 
gp: K = bnfinit(x^16 - 20*x^14 - 113*x^12 + 2010*x^10 + 9364*x^8 - 19600*x^6 - 80172*x^4 - 4890*x^2 + 72361, 1)
 

Normalized defining polynomial

\( x^{16} - 20 x^{14} - 113 x^{12} + 2010 x^{10} + 9364 x^{8} - 19600 x^{6} - 80172 x^{4} - 4890 x^{2} + 72361 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15413672017951201296000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.97$
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:  $2, 3, 5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{118} a^{12} + \frac{1}{118} a^{10} + \frac{19}{118} a^{8} - \frac{17}{118} a^{6} - \frac{35}{118} a^{4} - \frac{19}{59} a^{2} - \frac{13}{118}$, $\frac{1}{118} a^{13} + \frac{1}{118} a^{11} + \frac{19}{118} a^{9} - \frac{17}{118} a^{7} - \frac{35}{118} a^{5} - \frac{19}{59} a^{3} - \frac{13}{118} a$, $\frac{1}{648563464366522} a^{14} - \frac{2056780978103}{648563464366522} a^{12} - \frac{5776234159947}{324281732183261} a^{10} - \frac{12847276337891}{324281732183261} a^{8} - \frac{244389643883775}{648563464366522} a^{6} + \frac{11983450025733}{324281732183261} a^{4} - \frac{152013980399891}{324281732183261} a^{2} - \frac{23849322654897}{324281732183261}$, $\frac{1}{174463571914594418} a^{15} - \frac{606649840980793}{174463571914594418} a^{13} + \frac{4556153218587623}{87231785957297209} a^{11} - \frac{14187806383128232}{87231785957297209} a^{9} + \frac{2899494268130213}{174463571914594418} a^{7} + \frac{36534900574733688}{87231785957297209} a^{5} - \frac{12013030557543573}{87231785957297209} a^{3} - \frac{9713827184334374}{87231785957297209} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:  \( 807311327.563 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4).C_2^3$ (as 16T315):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $(C_2\times D_4).C_2^3$
Character table for $(C_2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{177}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{885}) \), \(\Q(\zeta_{15})^+\), 4.4.435125.1, \(\Q(\sqrt{5}, \sqrt{177})\), 8.8.15336035015625.1

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 R R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ 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
$2$2.8.8.10$x^{8} + 2 x^{6} + 8 x^{3} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
3Data not computed
5Data not computed
59Data not computed