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Properties

Label 16.8.68188413910...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{14}\cdot 11^{2}\cdot 19^{2}\cdot 29^{6}$
Root discriminant $97.64$
Ramified primes $2, 3, 5, 11, 19, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1086

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![918393025, 0, -78186900, 0, -227795875, 0, -12589250, 0, 2444730, 0, 57700, 0, -3570, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 - 3570*x^12 + 57700*x^10 + 2444730*x^8 - 12589250*x^6 - 227795875*x^4 - 78186900*x^2 + 918393025)
 
gp: K = bnfinit(x^16 - 20*x^14 - 3570*x^12 + 57700*x^10 + 2444730*x^8 - 12589250*x^6 - 227795875*x^4 - 78186900*x^2 + 918393025, 1)
 

Normalized defining polynomial

\( x^{16} - 20 x^{14} - 3570 x^{12} + 57700 x^{10} + 2444730 x^{8} - 12589250 x^{6} - 227795875 x^{4} - 78186900 x^{2} + 918393025 \)

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:  \(68188413910586864400000000000000=2^{16}\cdot 3^{8}\cdot 5^{14}\cdot 11^{2}\cdot 19^{2}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.64$
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:  $2, 3, 5, 11, 19, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{1430} a^{12} + \frac{1}{130} a^{10} - \frac{3}{715} a^{8} - \frac{111}{286} a^{6} + \frac{21}{143} a^{4} - \frac{1}{286} a^{2} + \frac{5}{13}$, $\frac{1}{1430} a^{13} + \frac{1}{130} a^{11} - \frac{3}{715} a^{9} - \frac{111}{286} a^{7} + \frac{21}{143} a^{5} - \frac{1}{286} a^{3} + \frac{5}{13} a$, $\frac{1}{45404981769690188602042821107230} a^{14} - \frac{14014746654412897804932100603}{45404981769690188602042821107230} a^{12} - \frac{965855667009314344169948819396}{22702490884845094301021410553615} a^{10} + \frac{93434843512234958110463107313}{3492690905360783738618678546710} a^{8} - \frac{76171315838896050086623253822}{238973588261527308431804321617} a^{6} - \frac{3254371572205940216422421451135}{9080996353938037720408564221446} a^{4} + \frac{2079847326192137869000145093368}{4540498176969018860204282110723} a^{2} - \frac{64445374894483832592028616}{749133505522029180037004143}$, $\frac{1}{45404981769690188602042821107230} a^{15} - \frac{14014746654412897804932100603}{45404981769690188602042821107230} a^{13} - \frac{965855667009314344169948819396}{22702490884845094301021410553615} a^{11} + \frac{93434843512234958110463107313}{3492690905360783738618678546710} a^{9} - \frac{76171315838896050086623253822}{238973588261527308431804321617} a^{7} - \frac{3254371572205940216422421451135}{9080996353938037720408564221446} a^{5} + \frac{2079847326192137869000145093368}{4540498176969018860204282110723} a^{3} - \frac{64445374894483832592028616}{749133505522029180037004143} a$

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:  \( 2086297710.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1086:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 97 conjugacy class representatives for t16n1086 are not computed
Character table for t16n1086 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.32625.1, \(\Q(\zeta_{15})^+\), 4.4.725.1, 8.8.1064390625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.8.7$x^{8} + 2 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.8.8$x^{8} + 4 x^{5} + 8 x^{2} + 48$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$