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Properties

Label 16.0.37228626555...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{10}\cdot 5^{10}\cdot 41^{6}\cdot 97^{8}$
Root discriminant $167.18$
Ramified primes $2, 5, 41, 97$
Class number $58$ (GRH)
Class group $[58]$ (GRH)
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5672327056, -6349974312, 7424247588, -2806521416, 1402119634, -205130504, 22643979, 1785594, -461910, 134109, 7085, 3947, -1435, 49, -14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 14*x^14 + 49*x^13 - 1435*x^12 + 3947*x^11 + 7085*x^10 + 134109*x^9 - 461910*x^8 + 1785594*x^7 + 22643979*x^6 - 205130504*x^5 + 1402119634*x^4 - 2806521416*x^3 + 7424247588*x^2 - 6349974312*x + 5672327056)
 
gp: K = bnfinit(x^16 - 2*x^15 - 14*x^14 + 49*x^13 - 1435*x^12 + 3947*x^11 + 7085*x^10 + 134109*x^9 - 461910*x^8 + 1785594*x^7 + 22643979*x^6 - 205130504*x^5 + 1402119634*x^4 - 2806521416*x^3 + 7424247588*x^2 - 6349974312*x + 5672327056, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 14 x^{14} + 49 x^{13} - 1435 x^{12} + 3947 x^{11} + 7085 x^{10} + 134109 x^{9} - 461910 x^{8} + 1785594 x^{7} + 22643979 x^{6} - 205130504 x^{5} + 1402119634 x^{4} - 2806521416 x^{3} + 7424247588 x^{2} - 6349974312 x + 5672327056 \)

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:  \(372286265552058761987916010000000000=2^{10}\cdot 5^{10}\cdot 41^{6}\cdot 97^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.18$
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:  $2, 5, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{820} a^{12} + \frac{1}{164} a^{11} + \frac{91}{410} a^{10} - \frac{36}{205} a^{9} + \frac{93}{820} a^{8} + \frac{5}{41} a^{7} + \frac{147}{410} a^{6} + \frac{75}{164} a^{5} + \frac{231}{820} a^{4} - \frac{91}{410} a^{3} - \frac{47}{205} a^{2} + \frac{9}{205} a - \frac{66}{205}$, $\frac{1}{820} a^{13} + \frac{157}{820} a^{11} + \frac{44}{205} a^{10} - \frac{7}{820} a^{9} + \frac{9}{164} a^{8} + \frac{51}{205} a^{7} + \frac{27}{164} a^{6} - \frac{1}{205} a^{5} - \frac{107}{820} a^{4} - \frac{49}{410} a^{3} + \frac{39}{205} a^{2} + \frac{94}{205} a - \frac{16}{41}$, $\frac{1}{30340} a^{14} + \frac{13}{30340} a^{13} + \frac{9}{30340} a^{12} - \frac{5493}{30340} a^{11} + \frac{973}{6068} a^{10} + \frac{99}{410} a^{9} + \frac{931}{6068} a^{8} + \frac{127}{740} a^{7} + \frac{14409}{30340} a^{6} - \frac{7279}{30340} a^{5} - \frac{8617}{30340} a^{4} + \frac{2252}{7585} a^{3} - \frac{3103}{7585} a^{2} + \frac{331}{1517} a + \frac{2988}{7585}$, $\frac{1}{1823169914989428964165846106578768850713150913526834051540507560} a^{15} - \frac{1413873422090546547289673960499980531174344067203798379817}{455792478747357241041461526644692212678287728381708512885126890} a^{14} + \frac{364762519182443320989371242943064691644055091754883436539279}{911584957494714482082923053289384425356575456763417025770253780} a^{13} - \frac{91792542171401008502057234468127715686876655721705637553}{49274862567281863896374219096723482451706781446671190582175880} a^{12} + \frac{155900855523828786513114363147762462393996148874295243007337057}{1823169914989428964165846106578768850713150913526834051540507560} a^{11} + \frac{552897641202733052310439489685375688524411699607781686037137}{364633982997885792833169221315753770142630182705366810308101512} a^{10} + \frac{444874483581434634350071047349098681062496949376305657285000031}{1823169914989428964165846106578768850713150913526834051540507560} a^{9} - \frac{29258805112481450497227105304965379550611860892667599206358823}{364633982997885792833169221315753770142630182705366810308101512} a^{8} - \frac{10808775100807569486127549767612469487285125417707817150472404}{45579247874735724104146152664469221267828772838170851288512689} a^{7} + \frac{437871079629982001654240240957040417560837169006222452229816243}{911584957494714482082923053289384425356575456763417025770253780} a^{6} + \frac{767342073190325575744889381824085013149567555626458837908573469}{1823169914989428964165846106578768850713150913526834051540507560} a^{5} - \frac{89636598899204655681913745781218406049664854219282265894180701}{227896239373678620520730763322346106339143864190854256442563445} a^{4} - \frac{164394557041306677997196295974704557142008169719505498418389849}{911584957494714482082923053289384425356575456763417025770253780} a^{3} - \frac{4283488529410550872676514548432987223163411212586348033889748}{227896239373678620520730763322346106339143864190854256442563445} a^{2} + \frac{12076005985142316741481492722389371903703076405905726647634519}{91158495749471448208292305328938442535657545676341702577025378} a + \frac{8726746282933843093697845613393854477234773465485734505885593}{20717839943061692774611887574758736939922169471895841494778495}$

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

Class group and class number

$C_{58}$, which has order $58$ (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:  \( 6751361707.36 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{97}) \), \(\Q(\sqrt{485}) \), 4.0.1025.1, 4.0.9644225.1, \(\Q(\sqrt{5}, \sqrt{97})\), 8.0.93011075850625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$