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Properties

Label 16.0.76087965578...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{12}\cdot 11^{6}$
Root discriminant $55.28$
Ramified primes $2, 5, 11$
Class number $800$ (GRH)
Class group $[2, 20, 20]$ (GRH)
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![153991, -673492, 1278140, -1295280, 1171864, -792892, 504548, -243740, 116061, -42732, 16008, -4236, 1224, -224, 44, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 44*x^14 - 224*x^13 + 1224*x^12 - 4236*x^11 + 16008*x^10 - 42732*x^9 + 116061*x^8 - 243740*x^7 + 504548*x^6 - 792892*x^5 + 1171864*x^4 - 1295280*x^3 + 1278140*x^2 - 673492*x + 153991)
 
gp: K = bnfinit(x^16 - 4*x^15 + 44*x^14 - 224*x^13 + 1224*x^12 - 4236*x^11 + 16008*x^10 - 42732*x^9 + 116061*x^8 - 243740*x^7 + 504548*x^6 - 792892*x^5 + 1171864*x^4 - 1295280*x^3 + 1278140*x^2 - 673492*x + 153991, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 44 x^{14} - 224 x^{13} + 1224 x^{12} - 4236 x^{11} + 16008 x^{10} - 42732 x^{9} + 116061 x^{8} - 243740 x^{7} + 504548 x^{6} - 792892 x^{5} + 1171864 x^{4} - 1295280 x^{3} + 1278140 x^{2} - 673492 x + 153991 \)

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:  \(7608796557869056000000000000=2^{44}\cdot 5^{12}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.28$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{68367805946675110161113635894796798885388461} a^{15} + \frac{22311134184944255892278725783787552876947203}{68367805946675110161113635894796798885388461} a^{14} - \frac{16917434794050577972571712159016508963967552}{68367805946675110161113635894796798885388461} a^{13} - \frac{2990440813933479666954954903665570823057036}{68367805946675110161113635894796798885388461} a^{12} + \frac{11390358390259034407606223127751866966059983}{68367805946675110161113635894796798885388461} a^{11} - \frac{909853631947167399323655898154499349619492}{68367805946675110161113635894796798885388461} a^{10} - \frac{14544083684970836266602806616629576670141856}{68367805946675110161113635894796798885388461} a^{9} + \frac{20356690031467831119138846372721396013877752}{68367805946675110161113635894796798885388461} a^{8} + \frac{831946928386576597869788422646996053429571}{68367805946675110161113635894796798885388461} a^{7} + \frac{32342621658897824007337550112854089803060180}{68367805946675110161113635894796798885388461} a^{6} - \frac{9098210643245111409357531165282757055492600}{68367805946675110161113635894796798885388461} a^{5} + \frac{20881298188215342492259280678015514071397011}{68367805946675110161113635894796798885388461} a^{4} - \frac{3064428354423072981215059154622865295315898}{68367805946675110161113635894796798885388461} a^{3} + \frac{23003846424859925278778621820530086854850853}{68367805946675110161113635894796798885388461} a^{2} - \frac{27063109756504972077089966632325047566369859}{68367805946675110161113635894796798885388461} a + \frac{21019748581472643680076289979458764578128941}{68367805946675110161113635894796798885388461}$

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

Class group and class number

$C_{2}\times C_{20}\times C_{20}$, which has order $800$ (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:  \( 17367.3059334 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.4400.1, 4.4.17600.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.4956160000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$