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Properties

Label 16.0.12882531132...3056.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 17^{8}$
Root discriminant $57.13$
Ramified primes $2, 3, 17$
Class number $1280$ (GRH)
Class group $[2, 4, 4, 40]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3186913, -716904, 2362076, -149280, 766944, 9744, 166628, 5952, 24297, 1008, 2624, -24, 218, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 12*x^14 + 218*x^12 - 24*x^11 + 2624*x^10 + 1008*x^9 + 24297*x^8 + 5952*x^7 + 166628*x^6 + 9744*x^5 + 766944*x^4 - 149280*x^3 + 2362076*x^2 - 716904*x + 3186913)
 
gp: K = bnfinit(x^16 + 12*x^14 + 218*x^12 - 24*x^11 + 2624*x^10 + 1008*x^9 + 24297*x^8 + 5952*x^7 + 166628*x^6 + 9744*x^5 + 766944*x^4 - 149280*x^3 + 2362076*x^2 - 716904*x + 3186913, 1)
 

Normalized defining polynomial

\( x^{16} + 12 x^{14} + 218 x^{12} - 24 x^{11} + 2624 x^{10} + 1008 x^{9} + 24297 x^{8} + 5952 x^{7} + 166628 x^{6} + 9744 x^{5} + 766944 x^{4} - 149280 x^{3} + 2362076 x^{2} - 716904 x + 3186913 \)

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:  \(12882531132048216201998893056=2^{48}\cdot 3^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.13$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(816=2^{4}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{816}(1,·)$, $\chi_{816}(67,·)$, $\chi_{816}(647,·)$, $\chi_{816}(713,·)$, $\chi_{816}(205,·)$, $\chi_{816}(271,·)$, $\chi_{816}(409,·)$, $\chi_{816}(475,·)$, $\chi_{816}(101,·)$, $\chi_{816}(35,·)$, $\chi_{816}(613,·)$, $\chi_{816}(679,·)$, $\chi_{816}(239,·)$, $\chi_{816}(305,·)$, $\chi_{816}(443,·)$, $\chi_{816}(509,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{8809727} a^{14} - \frac{515571}{8809727} a^{13} - \frac{1780669}{8809727} a^{12} + \frac{465788}{8809727} a^{11} - \frac{3514110}{8809727} a^{10} - \frac{3257963}{8809727} a^{9} + \frac{2563009}{8809727} a^{8} - \frac{753121}{8809727} a^{7} - \frac{1588824}{8809727} a^{6} - \frac{2811046}{8809727} a^{5} - \frac{151793}{8809727} a^{4} + \frac{3557897}{8809727} a^{3} - \frac{2502606}{8809727} a^{2} - \frac{1706825}{8809727} a - \frac{1567764}{8809727}$, $\frac{1}{67518876307669733372578121135183} a^{15} - \frac{3434129934068939491240398}{67518876307669733372578121135183} a^{14} + \frac{4700476695450309538141530330667}{67518876307669733372578121135183} a^{13} - \frac{14646644893914640891443342252732}{67518876307669733372578121135183} a^{12} - \frac{6626599781609529143875054331893}{67518876307669733372578121135183} a^{11} + \frac{15916927620249474937256625692933}{67518876307669733372578121135183} a^{10} + \frac{913551920394729407941189598834}{67518876307669733372578121135183} a^{9} - \frac{21418294284942479602935075818791}{67518876307669733372578121135183} a^{8} + \frac{24232777039024572103334367924983}{67518876307669733372578121135183} a^{7} + \frac{30285028034294112087582267856302}{67518876307669733372578121135183} a^{6} + \frac{5006500760817491514684088595972}{67518876307669733372578121135183} a^{5} - \frac{22642758398830109654431212827272}{67518876307669733372578121135183} a^{4} + \frac{16260824010201055070144699818477}{67518876307669733372578121135183} a^{3} - \frac{12523311713412837536258711611168}{67518876307669733372578121135183} a^{2} + \frac{12121159593800185397370597989866}{67518876307669733372578121135183} a + \frac{33566763468792853631900056427201}{67518876307669733372578121135183}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{40}$, which has order $1280$ (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:  \( 11964.310642723332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{-102}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{3}, \sqrt{-34})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(\sqrt{6}, \sqrt{-34})\), \(\Q(\sqrt{2}, \sqrt{-51})\), \(\Q(\sqrt{6}, \sqrt{-17})\), 4.0.591872.5, 4.0.5326848.5, \(\Q(\zeta_{16})^+\), 4.4.18432.1, 8.0.443364212736.5, 8.0.113501238460416.68, \(\Q(\zeta_{48})^+\), 8.0.1401249857536.2, 8.0.113501238460416.96, 8.0.28375309615104.64, 8.0.28375309615104.148

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$