Properties

Label 16.8.216...993.2
Degree $16$
Signature $[8, 4]$
Discriminant $2.165\times 10^{36}$
Root discriminant $186.62$
Ramified primes $43, 97$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 17*x^14 + 27*x^13 - 2446*x^12 - 10161*x^11 + 7582*x^10 + 42551*x^9 + 291322*x^8 + 12968232*x^7 + 37578055*x^6 - 154331401*x^5 - 321602800*x^4 - 85014601*x^3 + 505836265*x^2 + 5247291674*x - 5285070607)
 
gp: K = bnfinit(x^16 - 3*x^15 - 17*x^14 + 27*x^13 - 2446*x^12 - 10161*x^11 + 7582*x^10 + 42551*x^9 + 291322*x^8 + 12968232*x^7 + 37578055*x^6 - 154331401*x^5 - 321602800*x^4 - 85014601*x^3 + 505836265*x^2 + 5247291674*x - 5285070607, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5285070607, 5247291674, 505836265, -85014601, -321602800, -154331401, 37578055, 12968232, 291322, 42551, 7582, -10161, -2446, 27, -17, -3, 1]);
 

\( x^{16} - 3 x^{15} - 17 x^{14} + 27 x^{13} - 2446 x^{12} - 10161 x^{11} + 7582 x^{10} + 42551 x^{9} + 291322 x^{8} + 12968232 x^{7} + 37578055 x^{6} - 154331401 x^{5} - 321602800 x^{4} - 85014601 x^{3} + 505836265 x^{2} + 5247291674 x - 5285070607 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2164959798672044689794137876838502993\)\(\medspace = 43^{4}\cdot 97^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $186.62$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $43, 97$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{15} - \frac{153153548715593744200084925787938939664064903371082489381884942630717235115}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{14} - \frac{39375246780740927967002112480047278366661856807662675106690986443739286059}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{13} + \frac{111997492897976491827806514020362400503198330345099496068315524947458446444}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{12} - \frac{17357909374418113498795041158929515875840625496452021763441368420998309640}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{11} - \frac{66692702401028623898283719096787254028308782097766991427448968316736131926}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{10} - \frac{40065343531847748474976778142025061411283448868050353378721558904038469814}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{9} + \frac{184247914960436479415884475356461499455178567067615412154662529052949587411}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{8} - \frac{86171691771823038542870839607735243707483901626664949787038919813974603360}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{7} + \frac{87410749528085652435293952550496476130755987774566518324762679616126312422}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{6} + \frac{104983340108009359351024419558130931066425680250677188043315983378759463367}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{5} + \frac{93750281219974099606327715716217083766397252679882024976181654230102754899}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{4} - \frac{138678804906695447425635765929932985852084755008077209493795803424217147621}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{3} - \frac{135894138514331344209133304692886409255752555602187854038481506671360059431}{369160217966069108863874344047298585484513186045583714267969243733519321453} a^{2} + \frac{126607399860239133356623849793889988704480634646109141994725736409321871926}{369160217966069108863874344047298585484513186045583714267969243733519321453} a - \frac{157935171119815223699041525286594532373599267419558420984290726832865315845}{369160217966069108863874344047298585484513186045583714267969243733519321453}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 274195718818 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{4}\cdot 274195718818 \cdot 4}{2\sqrt{2164959798672044689794137876838502993}}\approx 0.148704868016256$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$43$43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.4.0.1$x^{4} - x + 20$$1$$4$$0$$C_4$$[\ ]^{4}$
43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97Data not computed