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Properties

Label 16.0.11023665190...3041.2
Degree $16$
Signature $[0, 8]$
Discriminant $11^{12}\cdot 37^{8}$
Root discriminant $36.74$
Ramified primes $11, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13225, 575, 20571, -9330, 21963, -10912, 16285, -9035, 6842, -3879, 1539, -630, 193, -64, 21, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 21*x^14 - 64*x^13 + 193*x^12 - 630*x^11 + 1539*x^10 - 3879*x^9 + 6842*x^8 - 9035*x^7 + 16285*x^6 - 10912*x^5 + 21963*x^4 - 9330*x^3 + 20571*x^2 + 575*x + 13225)
 
gp: K = bnfinit(x^16 - 5*x^15 + 21*x^14 - 64*x^13 + 193*x^12 - 630*x^11 + 1539*x^10 - 3879*x^9 + 6842*x^8 - 9035*x^7 + 16285*x^6 - 10912*x^5 + 21963*x^4 - 9330*x^3 + 20571*x^2 + 575*x + 13225, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 21 x^{14} - 64 x^{13} + 193 x^{12} - 630 x^{11} + 1539 x^{10} - 3879 x^{9} + 6842 x^{8} - 9035 x^{7} + 16285 x^{6} - 10912 x^{5} + 21963 x^{4} - 9330 x^{3} + 20571 x^{2} + 575 x + 13225 \)

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:  \(11023665190835148548573041=11^{12}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.74$
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:  $11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{3}{8} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{176} a^{12} + \frac{5}{176} a^{11} - \frac{5}{176} a^{10} - \frac{1}{22} a^{9} - \frac{5}{22} a^{8} - \frac{67}{176} a^{7} + \frac{13}{44} a^{6} + \frac{13}{176} a^{5} - \frac{39}{88} a^{4} + \frac{15}{88} a^{3} + \frac{67}{176} a^{2} - \frac{69}{176} a - \frac{13}{176}$, $\frac{1}{352} a^{13} + \frac{7}{176} a^{11} + \frac{17}{352} a^{10} - \frac{1}{8} a^{9} - \frac{175}{352} a^{8} + \frac{167}{352} a^{7} - \frac{159}{352} a^{6} - \frac{5}{32} a^{5} - \frac{19}{44} a^{4} + \frac{49}{352} a^{3} - \frac{3}{11} a^{2} - \frac{5}{88} a - \frac{155}{352}$, $\frac{1}{704} a^{14} - \frac{1}{704} a^{13} - \frac{1}{352} a^{12} + \frac{1}{64} a^{11} + \frac{19}{704} a^{10} - \frac{91}{704} a^{9} - \frac{169}{352} a^{8} + \frac{153}{352} a^{7} - \frac{25}{88} a^{6} - \frac{129}{704} a^{5} + \frac{305}{704} a^{4} - \frac{9}{704} a^{3} - \frac{7}{176} a^{2} - \frac{87}{704} a - \frac{7}{64}$, $\frac{1}{27839630250693840293714560} a^{15} - \frac{267689718213211303161}{1391981512534692014685728} a^{14} + \frac{207567842520906662021}{679015371968142446188160} a^{13} - \frac{52819105478062968382399}{27839630250693840293714560} a^{12} + \frac{55740336828130975405029}{13919815125346920146857280} a^{11} + \frac{3142490099984927743807}{63271886933395091576624} a^{10} - \frac{3432700232701977519340401}{27839630250693840293714560} a^{9} + \frac{2456616989896973350656989}{6959907562673460073428640} a^{8} + \frac{2493025791951202609923731}{13919815125346920146857280} a^{7} - \frac{608428677970550641223241}{5567926050138768058742912} a^{6} + \frac{4055885143625667236037}{347995378133673003671432} a^{5} + \frac{3460910262297208977144617}{6959907562673460073428640} a^{4} - \frac{9417409524327724265279877}{27839630250693840293714560} a^{3} - \frac{2421743358829037761236359}{5567926050138768058742912} a^{2} + \frac{57929857924261864277607}{316359434666975457883120} a + \frac{45389148829889302200209}{242083741310381219945344}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1185893.65744 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.4477.1, 8.0.741610573.1, 8.0.89734879333.1, 8.0.3320190535321.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$11$11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
37Data not computed