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Properties

Label 16.8.68470160614...3824.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{64}\cdot 3^{4}\cdot 7^{6}\cdot 79^{4}$
Root discriminant $130.23$
Ramified primes $2, 3, 7, 79$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T467)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![100457697148, 72795641824, -3013742336, -14285640128, -3246277392, 341051488, 217453872, 33122208, 3043444, -545552, -255872, -23232, -3048, -160, 72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 72*x^14 - 160*x^13 - 3048*x^12 - 23232*x^11 - 255872*x^10 - 545552*x^9 + 3043444*x^8 + 33122208*x^7 + 217453872*x^6 + 341051488*x^5 - 3246277392*x^4 - 14285640128*x^3 - 3013742336*x^2 + 72795641824*x + 100457697148)
 
gp: K = bnfinit(x^16 + 72*x^14 - 160*x^13 - 3048*x^12 - 23232*x^11 - 255872*x^10 - 545552*x^9 + 3043444*x^8 + 33122208*x^7 + 217453872*x^6 + 341051488*x^5 - 3246277392*x^4 - 14285640128*x^3 - 3013742336*x^2 + 72795641824*x + 100457697148, 1)
 

Normalized defining polynomial

\( x^{16} + 72 x^{14} - 160 x^{13} - 3048 x^{12} - 23232 x^{11} - 255872 x^{10} - 545552 x^{9} + 3043444 x^{8} + 33122208 x^{7} + 217453872 x^{6} + 341051488 x^{5} - 3246277392 x^{4} - 14285640128 x^{3} - 3013742336 x^{2} + 72795641824 x + 100457697148 \)

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:  \(6847016061481864788618324763213824=2^{64}\cdot 3^{4}\cdot 7^{6}\cdot 79^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.23$
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:  $2, 3, 7, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} + \frac{1}{12} a^{10} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{20842093655890991170689360662635650718678680652118613132538812115908791028} a^{15} - \frac{169007910479943832557282674735637014308547875753230165089362960494549353}{5210523413972747792672340165658912679669670163029653283134703028977197757} a^{14} - \frac{492763882107736324925354766159528887293356888414874907039316552188345623}{20842093655890991170689360662635650718678680652118613132538812115908791028} a^{13} + \frac{363036102884442684528796859873179442093252492261244866487960540804465359}{20842093655890991170689360662635650718678680652118613132538812115908791028} a^{12} - \frac{2213552498098667815267673666577641279828667931094249716247932887115190639}{20842093655890991170689360662635650718678680652118613132538812115908791028} a^{11} + \frac{273295340847843892763834189257943003718383949613561447832165730601438271}{6947364551963663723563120220878550239559560217372871044179604038636263676} a^{10} + \frac{332331060484467667539058269754694886291810545392180860080495099543037739}{10421046827945495585344680331317825359339340326059306566269406057954395514} a^{9} + \frac{22822006404060191490776865996945418685896208290890155354287122277928665}{3473682275981831861781560110439275119779780108686435522089802019318131838} a^{8} - \frac{429627206183614083473425458404555953418535935305471350446365271360954339}{1736841137990915930890780055219637559889890054343217761044901009659065919} a^{7} + \frac{744876201137987723915626213774455098491745058313390412965373246861314633}{3473682275981831861781560110439275119779780108686435522089802019318131838} a^{6} + \frac{63543351375340310561279787569350765854043440254135499076692309663375581}{3473682275981831861781560110439275119779780108686435522089802019318131838} a^{5} - \frac{1827895640254533246118886839549168501138448964519397585736343288734280517}{10421046827945495585344680331317825359339340326059306566269406057954395514} a^{4} - \frac{3378333559751782563580439403139921405444597688341510532717674919926748947}{10421046827945495585344680331317825359339340326059306566269406057954395514} a^{3} + \frac{56847700519048810225154164990107103096757119543705610692547668624646095}{3473682275981831861781560110439275119779780108686435522089802019318131838} a^{2} - \frac{290327098858349611147033518935702239444574456807349747007214042639231866}{5210523413972747792672340165658912679669670163029653283134703028977197757} a - \frac{1600760890918152857319566480719780177319110428165984850565626586212504}{4331274658331461174291222082841988927406209611828473219563344163738319}$

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:  \( 28102986463.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T467):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7168.1, \(\Q(\zeta_{16})^+\), 4.4.14336.1, 8.8.3288334336.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$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}$
79Data not computed