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Properties

Label 16.4.35847274805...0625.2
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 59^{8}$
Root discriminant $25.68$
Ramified primes $5, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -200, -200, 825, 180, -755, 380, -140, -364, -145, 98, 80, 19, -25, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 25*x^13 + 19*x^12 + 80*x^11 + 98*x^10 - 145*x^9 - 364*x^8 - 140*x^7 + 380*x^6 - 755*x^5 + 180*x^4 + 825*x^3 - 200*x^2 - 200*x + 25)
 
gp: K = bnfinit(x^16 - 8*x^14 - 25*x^13 + 19*x^12 + 80*x^11 + 98*x^10 - 145*x^9 - 364*x^8 - 140*x^7 + 380*x^6 - 755*x^5 + 180*x^4 + 825*x^3 - 200*x^2 - 200*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 25 x^{13} + 19 x^{12} + 80 x^{11} + 98 x^{10} - 145 x^{9} - 364 x^{8} - 140 x^{7} + 380 x^{6} - 755 x^{5} + 180 x^{4} + 825 x^{3} - 200 x^{2} - 200 x + 25 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(35847274805742431640625=5^{12}\cdot 59^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.68$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{30} a^{12} + \frac{1}{6} a^{11} - \frac{13}{30} a^{10} + \frac{7}{15} a^{8} + \frac{1}{6} a^{7} + \frac{4}{15} a^{6} - \frac{1}{2} a^{5} + \frac{1}{30} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{30} a^{13} - \frac{4}{15} a^{11} + \frac{1}{6} a^{10} + \frac{7}{15} a^{9} - \frac{1}{6} a^{8} + \frac{13}{30} a^{7} + \frac{1}{6} a^{6} - \frac{7}{15} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{30} a^{14} - \frac{1}{2} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{2}{5} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{60282064962010395930} a^{15} - \frac{79018608795768049}{30141032481005197965} a^{14} - \frac{500595917580923057}{30141032481005197965} a^{13} + \frac{557715066029582713}{60282064962010395930} a^{12} - \frac{270374259983833099}{30141032481005197965} a^{11} - \frac{28794742774944744329}{60282064962010395930} a^{10} - \frac{7979726487079956731}{60282064962010395930} a^{9} + \frac{4306924745633552949}{20094021654003465310} a^{8} - \frac{21402517188891577}{10047010827001732655} a^{7} + \frac{3351559144345112339}{60282064962010395930} a^{6} + \frac{3714202839393773737}{30141032481005197965} a^{5} - \frac{2531441533509564386}{10047010827001732655} a^{4} - \frac{3287639192210688095}{12056412992402079186} a^{3} - \frac{1592611331642495819}{4018804330800693062} a^{2} + \frac{2469088738272821042}{6028206496201039593} a + \frac{2986304652835913908}{6028206496201039593}$

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:  $9$
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:  \( 60268.4261403 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1475.1, 4.4.435125.1, 4.2.7375.1, 8.2.3209046875.2, 8.2.128361875.1, 8.4.189333765625.1

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 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ 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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
59Data not computed