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Properties

Label 16.0.20882706457...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 13^{8}$
Root discriminant $16.12$
Ramified primes $2, 5, 13$
Class number $2$
Class group $[2]$
Galois group $C_8:C_2^2$ (as 16T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 80, 239, 466, 632, 678, 495, 182, -89, -170, -49, 44, 34, 2, -7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 7*x^14 + 2*x^13 + 34*x^12 + 44*x^11 - 49*x^10 - 170*x^9 - 89*x^8 + 182*x^7 + 495*x^6 + 678*x^5 + 632*x^4 + 466*x^3 + 239*x^2 + 80*x + 25)
 
gp: K = bnfinit(x^16 - 2*x^15 - 7*x^14 + 2*x^13 + 34*x^12 + 44*x^11 - 49*x^10 - 170*x^9 - 89*x^8 + 182*x^7 + 495*x^6 + 678*x^5 + 632*x^4 + 466*x^3 + 239*x^2 + 80*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 7 x^{14} + 2 x^{13} + 34 x^{12} + 44 x^{11} - 49 x^{10} - 170 x^{9} - 89 x^{8} + 182 x^{7} + 495 x^{6} + 678 x^{5} + 632 x^{4} + 466 x^{3} + 239 x^{2} + 80 x + 25 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(20882706457600000000=2^{16}\cdot 5^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.12$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{3421741196237825} a^{15} + \frac{109653959092276}{3421741196237825} a^{14} - \frac{162235327862719}{3421741196237825} a^{13} + \frac{15861761626939}{684348239247565} a^{12} + \frac{1073330784647724}{3421741196237825} a^{11} - \frac{982006208780329}{3421741196237825} a^{10} + \frac{646832047161464}{3421741196237825} a^{9} - \frac{185748002386763}{3421741196237825} a^{8} + \frac{826686994558162}{3421741196237825} a^{7} + \frac{184656635373998}{3421741196237825} a^{6} - \frac{182719241953816}{3421741196237825} a^{5} - \frac{131845871712892}{684348239247565} a^{4} + \frac{498307137760397}{3421741196237825} a^{3} - \frac{1466901317796248}{3421741196237825} a^{2} - \frac{224206559606394}{684348239247565} a + \frac{281012780314}{136869647849513}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{711950936}{9891569549} a^{15} - \frac{2373243956}{9891569549} a^{14} - \frac{2399159426}{9891569549} a^{13} + \frac{6170622621}{9891569549} a^{12} + \frac{19128547115}{9891569549} a^{11} + \frac{2344397721}{9891569549} a^{10} - \frac{56461120219}{9891569549} a^{9} - \frac{58645796739}{9891569549} a^{8} + \frac{56599623083}{9891569549} a^{7} + \frac{130992075612}{9891569549} a^{6} + \frac{171369952884}{9891569549} a^{5} + \frac{127375501459}{9891569549} a^{4} + \frac{63089914860}{9891569549} a^{3} + \frac{27644725614}{9891569549} a^{2} - \frac{13925111620}{9891569549} a + \frac{514980568}{9891569549} \) (order $4$)
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:  \( 1959.48860283 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-65}) \), 4.2.16900.1 x2, 4.0.1040.1 x2, \(\Q(i, \sqrt{65})\), 8.0.70304000.2 x2, 8.0.70304000.1 x2, 8.0.4569760000.4

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 8 siblings: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$