↑

Properties

Label 16.8.35726164858...8976.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 7^{8}\cdot 137^{4}$
Root discriminant $60.89$
Ramified primes $2, 7, 137$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2.C_2\wr C_2^2$ (as 16T394)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-103, 3732, -29896, 5244, 50116, -90488, 20768, -31284, 3373, 5176, 604, 1252, -654, -68, 44, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 44*x^14 - 68*x^13 - 654*x^12 + 1252*x^11 + 604*x^10 + 5176*x^9 + 3373*x^8 - 31284*x^7 + 20768*x^6 - 90488*x^5 + 50116*x^4 + 5244*x^3 - 29896*x^2 + 3732*x - 103)
 
gp: K = bnfinit(x^16 - 8*x^15 + 44*x^14 - 68*x^13 - 654*x^12 + 1252*x^11 + 604*x^10 + 5176*x^9 + 3373*x^8 - 31284*x^7 + 20768*x^6 - 90488*x^5 + 50116*x^4 + 5244*x^3 - 29896*x^2 + 3732*x - 103, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 44 x^{14} - 68 x^{13} - 654 x^{12} + 1252 x^{11} + 604 x^{10} + 5176 x^{9} + 3373 x^{8} - 31284 x^{7} + 20768 x^{6} - 90488 x^{5} + 50116 x^{4} + 5244 x^{3} - 29896 x^{2} + 3732 x - 103 \)

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:  \(35726164858960310037146238976=2^{44}\cdot 7^{8}\cdot 137^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.89$
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:  $2, 7, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{213} a^{14} + \frac{1}{71} a^{13} - \frac{14}{213} a^{12} + \frac{2}{213} a^{11} - \frac{68}{213} a^{10} - \frac{104}{213} a^{9} - \frac{32}{213} a^{8} + \frac{17}{213} a^{7} + \frac{11}{213} a^{6} + \frac{7}{71} a^{5} - \frac{95}{213} a^{4} - \frac{8}{213} a^{3} + \frac{98}{213} a^{2} + \frac{7}{71} a + \frac{41}{213}$, $\frac{1}{683956475511950990916495586728351885183} a^{15} - \frac{124749001663942785771563147811748719}{227985491837316996972165195576117295061} a^{14} - \frac{45416156847740924616108556280761019774}{683956475511950990916495586728351885183} a^{13} - \frac{16931456890897932526427852812105220927}{227985491837316996972165195576117295061} a^{12} - \frac{30755136433321344452287982147522953813}{227985491837316996972165195576117295061} a^{11} + \frac{15657920292157562730798674145699194230}{683956475511950990916495586728351885183} a^{10} - \frac{245029146581867879891969347093561667080}{683956475511950990916495586728351885183} a^{9} - \frac{280270245153685007236714690231038118828}{683956475511950990916495586728351885183} a^{8} + \frac{133570520593618054861543517736557535019}{683956475511950990916495586728351885183} a^{7} + \frac{255463145306344515526928015056160362658}{683956475511950990916495586728351885183} a^{6} - \frac{72352336591576314263082024061707473442}{227985491837316996972165195576117295061} a^{5} + \frac{92068870640855094074976918257858947593}{227985491837316996972165195576117295061} a^{4} - \frac{65038318714913572318054415908137936916}{683956475511950990916495586728351885183} a^{3} + \frac{95477480919249112155446097870525246324}{227985491837316996972165195576117295061} a^{2} - \frac{86863604939625728067274163299159203531}{227985491837316996972165195576117295061} a - \frac{291770980099452395481588419699348036606}{683956475511950990916495586728351885183}$

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:  $11$
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:  \( 171364036.152 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.C_2\wr C_2^2$ (as 16T394):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 17 conjugacy class representatives for $C_2.C_2\wr C_2^2$
Character table for $C_2.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{14}) \), 4.4.7168.1 x2, 4.4.25088.1 x2, \(\Q(\sqrt{2}, \sqrt{7})\), 8.8.10070523904.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
137Data not computed