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Properties

Label 16.0.44157307385...8097.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 97^{13}$
Root discriminant $71.25$
Ramified primes $3, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14591551, -29838288, 20600666, -3332401, -1198369, -621919, 1010979, -385606, 129678, -60643, 19159, -3750, 1002, -256, 33, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 33*x^14 - 256*x^13 + 1002*x^12 - 3750*x^11 + 19159*x^10 - 60643*x^9 + 129678*x^8 - 385606*x^7 + 1010979*x^6 - 621919*x^5 - 1198369*x^4 - 3332401*x^3 + 20600666*x^2 - 29838288*x + 14591551)
 
gp: K = bnfinit(x^16 - 4*x^15 + 33*x^14 - 256*x^13 + 1002*x^12 - 3750*x^11 + 19159*x^10 - 60643*x^9 + 129678*x^8 - 385606*x^7 + 1010979*x^6 - 621919*x^5 - 1198369*x^4 - 3332401*x^3 + 20600666*x^2 - 29838288*x + 14591551, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 33 x^{14} - 256 x^{13} + 1002 x^{12} - 3750 x^{11} + 19159 x^{10} - 60643 x^{9} + 129678 x^{8} - 385606 x^{7} + 1010979 x^{6} - 621919 x^{5} - 1198369 x^{4} - 3332401 x^{3} + 20600666 x^{2} - 29838288 x + 14591551 \)

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:  \(441573073857633665833769108097=3^{8}\cdot 97^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.25$
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:  $3, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{103} a^{14} - \frac{1}{103} a^{13} + \frac{2}{103} a^{12} - \frac{16}{103} a^{11} - \frac{29}{103} a^{10} + \frac{10}{103} a^{9} + \frac{19}{103} a^{8} + \frac{7}{103} a^{7} + \frac{5}{103} a^{6} + \frac{51}{103} a^{5} + \frac{47}{103} a^{4} + \frac{47}{103} a^{3} - \frac{6}{103} a^{2} - \frac{37}{103} a + \frac{2}{103}$, $\frac{1}{1019715447364976396034706663824338211444440384380979} a^{15} - \frac{3690099416508988598000839925632973275118549628306}{1019715447364976396034706663824338211444440384380979} a^{14} + \frac{197843417278220553504887959820958342998402707917500}{1019715447364976396034706663824338211444440384380979} a^{13} - \frac{127577271670718735664876606811285796618750185511382}{1019715447364976396034706663824338211444440384380979} a^{12} + \frac{364998511794004194261642074814335397803598102823317}{1019715447364976396034706663824338211444440384380979} a^{11} - \frac{172447592304266969203744039325672856684111542911666}{1019715447364976396034706663824338211444440384380979} a^{10} - \frac{22987240562156567990246584430375300361837748266970}{78439649797305876618054358755718323957264644952383} a^{9} + \frac{182765214866086137592993432741208715763216083681502}{1019715447364976396034706663824338211444440384380979} a^{8} - \frac{373471306627120230859235481109878137323219676225255}{1019715447364976396034706663824338211444440384380979} a^{7} - \frac{230254873167913858093671762455574345293981141460784}{1019715447364976396034706663824338211444440384380979} a^{6} + \frac{91249832574475451640911964350553434120917907580775}{1019715447364976396034706663824338211444440384380979} a^{5} - \frac{209878800693158297801784348822642276833551532236091}{1019715447364976396034706663824338211444440384380979} a^{4} - \frac{424261504186440697622460407904344749546397402130636}{1019715447364976396034706663824338211444440384380979} a^{3} - \frac{340091528949940178950754189563332635211374374010284}{1019715447364976396034706663824338211444440384380979} a^{2} + \frac{246206933596462209915481867849414869894035056101336}{1019715447364976396034706663824338211444440384380979} a + \frac{6264971317582434261891854204861151361829004715357}{78439649797305876618054358755718323957264644952383}$

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

Class group and class number

$C_{4}$, 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:  \( \frac{73765450669752614272628988}{124340609372761150192765954666289} a^{15} - \frac{240893508237771447774855152}{124340609372761150192765954666289} a^{14} + \frac{2173605576895009187417460473}{124340609372761150192765954666289} a^{13} - \frac{17196970752061612691051408232}{124340609372761150192765954666289} a^{12} + \frac{59133330734452939663188136874}{124340609372761150192765954666289} a^{11} - \frac{218317403959240283104627291120}{124340609372761150192765954666289} a^{10} + \frac{1223100456730333835433716015985}{124340609372761150192765954666289} a^{9} - \frac{3406437740278623250086637425365}{124340609372761150192765954666289} a^{8} + \frac{6116819097393548069741472151569}{124340609372761150192765954666289} a^{7} - \frac{22536009843845240676880328064903}{124340609372761150192765954666289} a^{6} + \frac{55606321868310804099954857824481}{124340609372761150192765954666289} a^{5} + \frac{11503300957929471530479821982423}{124340609372761150192765954666289} a^{4} - \frac{91461475927673429745630113635820}{124340609372761150192765954666289} a^{3} - \frac{381725118617995318166719191446540}{124340609372761150192765954666289} a^{2} + \frac{1235492314328506477844616208029726}{124340609372761150192765954666289} a - \frac{849512977891019840946334934696650}{124340609372761150192765954666289} \) (order $6$)
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:  \( 103534343.224 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.873.1, 8.0.695574560817.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
3Data not computed
$97$97.8.6.4$x^{8} + 1358 x^{4} + 1176125$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
97.8.7.5$x^{8} + 485$$8$$1$$7$$C_8$$[\ ]_{8}$