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Properties

Label 16.0.11017296355...6624.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 13^{6}$
Root discriminant $23.86$
Ramified primes $2, 3, 13$
Class number $2$
Class group $[2]$
Galois group $SD_{16}:C_2$ (as 16T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![676, 1352, 832, 104, 816, 2312, 4072, 4824, 4402, 2964, 1436, 420, 32, -32, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 - 32*x^13 + 32*x^12 + 420*x^11 + 1436*x^10 + 2964*x^9 + 4402*x^8 + 4824*x^7 + 4072*x^6 + 2312*x^5 + 816*x^4 + 104*x^3 + 832*x^2 + 1352*x + 676)
 
gp: K = bnfinit(x^16 - 10*x^14 - 32*x^13 + 32*x^12 + 420*x^11 + 1436*x^10 + 2964*x^9 + 4402*x^8 + 4824*x^7 + 4072*x^6 + 2312*x^5 + 816*x^4 + 104*x^3 + 832*x^2 + 1352*x + 676, 1)
 

Normalized defining polynomial

\( x^{16} - 10 x^{14} - 32 x^{13} + 32 x^{12} + 420 x^{11} + 1436 x^{10} + 2964 x^{9} + 4402 x^{8} + 4824 x^{7} + 4072 x^{6} + 2312 x^{5} + 816 x^{4} + 104 x^{3} + 832 x^{2} + 1352 x + 676 \)

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:  \(11017296355467800346624=2^{32}\cdot 3^{12}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.86$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{78} a^{12} + \frac{1}{78} a^{11} - \frac{1}{13} a^{10} + \frac{17}{78} a^{9} - \frac{4}{39} a^{8} - \frac{3}{13} a^{7} + \frac{8}{39} a^{6} - \frac{2}{13} a^{5} + \frac{3}{13} a^{4} + \frac{11}{39} a^{3} - \frac{19}{39} a^{2} - \frac{1}{3}$, $\frac{1}{78} a^{13} - \frac{7}{78} a^{11} - \frac{8}{39} a^{10} + \frac{7}{39} a^{9} - \frac{5}{39} a^{8} + \frac{17}{39} a^{7} - \frac{14}{39} a^{6} + \frac{5}{13} a^{5} + \frac{2}{39} a^{4} + \frac{3}{13} a^{3} + \frac{19}{39} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{78} a^{14} - \frac{3}{26} a^{11} + \frac{11}{78} a^{10} - \frac{4}{39} a^{9} + \frac{17}{78} a^{8} + \frac{1}{39} a^{7} - \frac{7}{39} a^{6} - \frac{1}{39} a^{5} - \frac{2}{13} a^{4} + \frac{6}{13} a^{3} + \frac{10}{39} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{904639573744920858} a^{15} + \frac{114283752364093}{34793829759420033} a^{14} + \frac{42957778892261}{452319786872460429} a^{13} - \frac{1597214256846320}{452319786872460429} a^{12} + \frac{53237267956501167}{301546524581640286} a^{11} + \frac{195068949019474219}{904639573744920858} a^{10} + \frac{7175749448920651}{69587659518840066} a^{9} - \frac{4646344937131138}{34793829759420033} a^{8} + \frac{25048707385760009}{150773262290820143} a^{7} - \frac{124248690738636844}{452319786872460429} a^{6} - \frac{9733845110221070}{452319786872460429} a^{5} - \frac{77768309099883256}{452319786872460429} a^{4} + \frac{27458195767668074}{452319786872460429} a^{3} - \frac{152060810683207201}{452319786872460429} a^{2} + \frac{1013514886163192}{11597943253140011} a - \frac{1932273672247155}{11597943253140011}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{3872749921063}{1212653584108473} a^{15} + \frac{2241044372891}{1212653584108473} a^{14} + \frac{13345684332897}{404217861369491} a^{13} + \frac{32105477996022}{404217861369491} a^{12} - \frac{204233523192953}{1212653584108473} a^{11} - \frac{513630736092135}{404217861369491} a^{10} - \frac{1488962475216973}{404217861369491} a^{9} - \frac{16125339557646119}{2425307168216946} a^{8} - \frac{3518989162969142}{404217861369491} a^{7} - \frac{773595289925255}{93281044931421} a^{6} - \frac{7087419619409098}{1212653584108473} a^{5} - \frac{872829752695375}{404217861369491} a^{4} - \frac{156570579256640}{1212653584108473} a^{3} - \frac{178491231124942}{1212653584108473} a^{2} - \frac{52579971754152}{31093681643807} a - \frac{182007873403405}{93281044931421} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 142361.959547 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}:C_2$ (as 16T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $SD_{16}:C_2$
Character table for $SD_{16}:C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.0.7488.1, 4.4.7488.1, \(\Q(\zeta_{12})\), 8.0.56070144.2

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$