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Properties

Label 16.0.83609126117...9296.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 13^{8}\cdot 17^{12}$
Root discriminant $203.07$
Ramified primes $2, 13, 17$
Class number $13994240$ (GRH)
Class group $[2, 2, 2, 2, 116, 7540]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![58736635127, -4833136668, 20206455030, -1919250704, 2957550889, -336400104, 264514286, -32228676, 17396206, -1624572, 690830, -44888, 15581, -656, 190, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 190*x^14 - 656*x^13 + 15581*x^12 - 44888*x^11 + 690830*x^10 - 1624572*x^9 + 17396206*x^8 - 32228676*x^7 + 264514286*x^6 - 336400104*x^5 + 2957550889*x^4 - 1919250704*x^3 + 20206455030*x^2 - 4833136668*x + 58736635127)
 
gp: K = bnfinit(x^16 - 4*x^15 + 190*x^14 - 656*x^13 + 15581*x^12 - 44888*x^11 + 690830*x^10 - 1624572*x^9 + 17396206*x^8 - 32228676*x^7 + 264514286*x^6 - 336400104*x^5 + 2957550889*x^4 - 1919250704*x^3 + 20206455030*x^2 - 4833136668*x + 58736635127, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 190 x^{14} - 656 x^{13} + 15581 x^{12} - 44888 x^{11} + 690830 x^{10} - 1624572 x^{9} + 17396206 x^{8} - 32228676 x^{7} + 264514286 x^{6} - 336400104 x^{5} + 2957550889 x^{4} - 1919250704 x^{3} + 20206455030 x^{2} - 4833136668 x + 58736635127 \)

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:  \(8360912611710390029718223769930039296=2^{44}\cdot 13^{8}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.07$
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:  $2, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3536=2^{4}\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3536}(1665,·)$, $\chi_{3536}(259,·)$, $\chi_{3536}(1,·)$, $\chi_{3536}(1611,·)$, $\chi_{3536}(1041,·)$, $\chi_{3536}(2755,·)$, $\chi_{3536}(3379,·)$, $\chi_{3536}(2393,·)$, $\chi_{3536}(2651,·)$, $\chi_{3536}(987,·)$, $\chi_{3536}(3433,·)$, $\chi_{3536}(2027,·)$, $\chi_{3536}(625,·)$, $\chi_{3536}(883,·)$, $\chi_{3536}(1769,·)$, $\chi_{3536}(2809,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13} a^{8} - \frac{2}{13} a^{7} + \frac{2}{13} a^{6} + \frac{1}{13} a^{5} - \frac{3}{13} a^{4} - \frac{1}{13} a^{3} + \frac{2}{13} a^{2} + \frac{2}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{9} - \frac{2}{13} a^{7} + \frac{5}{13} a^{6} - \frac{1}{13} a^{5} + \frac{6}{13} a^{4} + \frac{6}{13} a^{2} + \frac{5}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{10} + \frac{1}{13} a^{7} + \frac{3}{13} a^{6} - \frac{5}{13} a^{5} - \frac{6}{13} a^{4} + \frac{4}{13} a^{3} - \frac{4}{13} a^{2} + \frac{6}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{11} + \frac{5}{13} a^{7} + \frac{6}{13} a^{6} + \frac{6}{13} a^{5} - \frac{6}{13} a^{4} - \frac{3}{13} a^{3} + \frac{4}{13} a^{2} - \frac{1}{13}$, $\frac{1}{104} a^{12} - \frac{1}{26} a^{10} - \frac{1}{26} a^{9} - \frac{1}{52} a^{8} + \frac{3}{13} a^{7} + \frac{19}{52} a^{6} + \frac{3}{13} a^{5} - \frac{1}{13} a^{4} - \frac{11}{26} a^{3} + \frac{1}{26} a^{2} + \frac{4}{13} a - \frac{23}{104}$, $\frac{1}{104} a^{13} - \frac{1}{26} a^{11} - \frac{1}{26} a^{10} - \frac{1}{52} a^{9} - \frac{9}{52} a^{7} - \frac{3}{13} a^{6} - \frac{4}{13} a^{5} + \frac{7}{26} a^{4} + \frac{7}{26} a^{3} - \frac{2}{13} a^{2} + \frac{33}{104} a - \frac{3}{13}$, $\frac{1}{43898777277681402707280} a^{14} + \frac{21953186114097612241}{10974694319420350676820} a^{13} + \frac{133350996106103249083}{43898777277681402707280} a^{12} + \frac{2283140447433566789}{233504134455752142060} a^{11} + \frac{102733180689271558529}{4389877727768140270728} a^{10} - \frac{28133825097913982093}{3658231439806783558940} a^{9} - \frac{173010262609436492161}{5487347159710175338410} a^{8} - \frac{495494455191325740322}{2743673579855087669205} a^{7} + \frac{1655933617325284884367}{7316462879613567117880} a^{6} - \frac{3835552696943081383273}{10974694319420350676820} a^{5} + \frac{156727131679657721111}{3658231439806783558940} a^{4} - \frac{18915185822228319745}{46700826891150428412} a^{3} + \frac{9885051788673211948301}{43898777277681402707280} a^{2} + \frac{421028596101014363881}{10974694319420350676820} a - \frac{205955974488535323271}{934016537823008568240}$, $\frac{1}{1580903554755630537798724995242989406702640} a^{15} - \frac{11726872603088149889}{1580903554755630537798724995242989406702640} a^{14} - \frac{2337149149402560271321108487725362954389}{1580903554755630537798724995242989406702640} a^{13} + \frac{2591284530289163305206272112627909827113}{1580903554755630537798724995242989406702640} a^{12} - \frac{9182792055584889556498149666812518639433}{790451777377815268899362497621494703351320} a^{11} - \frac{38722822353505545485585626238712132549}{2960493548231517861046301489219081285960} a^{10} + \frac{1306299699270021405073980115789225518659}{79045177737781526889936249762149470335132} a^{9} - \frac{1039994528482865641692047054288449210843}{98806472172226908612420312202686837918915} a^{8} - \frac{12204128481277588563978619876598420948517}{263483925792605089633120832540498234450440} a^{7} - \frac{148802773902719438151133203958334292558119}{790451777377815268899362497621494703351320} a^{6} + \frac{14384652548618796203362488447009375927447}{65870981448151272408280208135124558612610} a^{5} - \frac{20364939545571630610856395445910312141271}{98806472172226908612420312202686837918915} a^{4} + \frac{732985194840630183255288430369044053104361}{1580903554755630537798724995242989406702640} a^{3} - \frac{743889470552374482526062378555480709729469}{1580903554755630537798724995242989406702640} a^{2} + \frac{211616753523157694963432791050513246486151}{1580903554755630537798724995242989406702640} a + \frac{3452785728271631377339376004619172706741}{11212081948621493175877482235765882317040}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{116}\times C_{7540}$, which has order $13994240$ (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:  \( 103646.40189541418 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.0.100026368.5, \(\Q(\sqrt{2}, \sqrt{17})\), 4.0.346112.2, 4.4.314432.1, 4.4.4913.1, 4.0.1700448256.3, 4.0.1700448256.4, 8.0.10005274295271424.160, 8.8.98867482624.1, 8.0.2891524271333441536.6

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$