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Properties

Label 16.0.45315174505...7616.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 7^{6}\cdot 17^{4}$
Root discriminant $61.80$
Ramified primes $2, 7, 17$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42191, -137400, 129552, 70808, -233104, 155224, 34688, -110584, 65694, -10936, -5672, 3224, -528, -24, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 - 24*x^13 - 528*x^12 + 3224*x^11 - 5672*x^10 - 10936*x^9 + 65694*x^8 - 110584*x^7 + 34688*x^6 + 155224*x^5 - 233104*x^4 + 70808*x^3 + 129552*x^2 - 137400*x + 42191)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 - 24*x^13 - 528*x^12 + 3224*x^11 - 5672*x^10 - 10936*x^9 + 65694*x^8 - 110584*x^7 + 34688*x^6 + 155224*x^5 - 233104*x^4 + 70808*x^3 + 129552*x^2 - 137400*x + 42191, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} - 24 x^{13} - 528 x^{12} + 3224 x^{11} - 5672 x^{10} - 10936 x^{9} + 65694 x^{8} - 110584 x^{7} + 34688 x^{6} + 155224 x^{5} - 233104 x^{4} + 70808 x^{3} + 129552 x^{2} - 137400 x + 42191 \)

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:  \(45315174505109995158677487616=2^{62}\cdot 7^{6}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.80$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} + \frac{8}{31} a^{13} + \frac{11}{31} a^{12} + \frac{12}{31} a^{11} + \frac{14}{31} a^{10} + \frac{14}{31} a^{9} + \frac{11}{31} a^{8} + \frac{14}{31} a^{6} + \frac{2}{31} a^{4} + \frac{8}{31} a^{3} - \frac{15}{31} a^{2} - \frac{4}{31} a$, $\frac{1}{93327561494102262165344740773311268833} a^{15} + \frac{30816179960924743418975939849429441}{3010566499809750392430475508816492543} a^{14} - \frac{33171786739183657269550063368930086418}{93327561494102262165344740773311268833} a^{13} + \frac{34899898409719069561368478468451960383}{93327561494102262165344740773311268833} a^{12} - \frac{45057910778358086123413505973914979680}{93327561494102262165344740773311268833} a^{11} - \frac{10014075572956236191187478782123993937}{93327561494102262165344740773311268833} a^{10} - \frac{43663128807771856160798043937484360857}{93327561494102262165344740773311268833} a^{9} - \frac{5525361303633445669526675448863408929}{93327561494102262165344740773311268833} a^{8} + \frac{23229679838316962274150991292300212294}{93327561494102262165344740773311268833} a^{7} + \frac{35626705889902074326613812118318659051}{93327561494102262165344740773311268833} a^{6} - \frac{35806871146784394583572393333225233327}{93327561494102262165344740773311268833} a^{5} + \frac{27735020531724158179770122788931504424}{93327561494102262165344740773311268833} a^{4} + \frac{45483713897659726673863833495475509611}{93327561494102262165344740773311268833} a^{3} - \frac{22916123070729992738199756258909945468}{93327561494102262165344740773311268833} a^{2} + \frac{22763562972959810052215098943220655465}{93327561494102262165344740773311268833} a - \frac{726992909510341948354681954108362466}{3010566499809750392430475508816492543}$

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:  $7$
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:  \( 18103071.8835 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.0.7168.1, 4.0.14336.1, 8.0.3288334336.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{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$$[\ ]_{2}^{4}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$