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Properties

Label 16.4.16240712178...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{36}\cdot 5^{14}\cdot 61^{2}\cdot 101^{4}$
Root discriminant $103.08$
Ramified primes $2, 5, 61, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![948948025, 0, -284022100, 0, -55698400, 0, -10099500, 0, -1146490, 0, -37820, 0, 880, 0, 60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 60*x^14 + 880*x^12 - 37820*x^10 - 1146490*x^8 - 10099500*x^6 - 55698400*x^4 - 284022100*x^2 + 948948025)
 
gp: K = bnfinit(x^16 + 60*x^14 + 880*x^12 - 37820*x^10 - 1146490*x^8 - 10099500*x^6 - 55698400*x^4 - 284022100*x^2 + 948948025, 1)
 

Normalized defining polynomial

\( x^{16} + 60 x^{14} + 880 x^{12} - 37820 x^{10} - 1146490 x^{8} - 10099500 x^{6} - 55698400 x^{4} - 284022100 x^{2} + 948948025 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(162407121785611878400000000000000=2^{36}\cdot 5^{14}\cdot 61^{2}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.08$
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:  $2, 5, 61, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{8} - \frac{1}{4}$, $\frac{1}{40} a^{9} - \frac{1}{40} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{40} a^{10} - \frac{1}{40} a^{8} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{40} a^{11} - \frac{1}{40} a^{8} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{80} a^{13} - \frac{1}{80} a^{12} - \frac{1}{80} a^{9} + \frac{1}{80} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{3}{16} a^{5} - \frac{3}{16} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{3}{16} a + \frac{3}{16}$, $\frac{1}{46074143220770064820352184400} a^{14} - \frac{62039289895299780236582369}{9214828644154012964070436880} a^{12} + \frac{1335346919385584983694837}{224751918150097877172449680} a^{10} - \frac{405790629427624357191673}{151062764658262507607712080} a^{8} + \frac{2176089505123653838585082911}{9214828644154012964070436880} a^{6} - \frac{267125083994868793146852375}{1842965728830802592814087376} a^{4} - \frac{34558325681213220912961071}{96998196254252768042846704} a^{2} - \frac{53912588877881247570195}{299134187442103975460816}$, $\frac{1}{46074143220770064820352184400} a^{15} + \frac{13286517039156345453574523}{2303707161038503241017609220} a^{13} - \frac{1}{80} a^{12} + \frac{1335346919385584983694837}{224751918150097877172449680} a^{11} + \frac{185311741100082123488091}{18882845582282813450964010} a^{9} - \frac{1}{80} a^{8} - \frac{127617655914849402432526309}{9214828644154012964070436880} a^{7} - \frac{1}{4} a^{6} + \frac{4901936885056668312861813}{115185358051925162050880461} a^{5} - \frac{3}{16} a^{4} + \frac{38190321509476355119173957}{96998196254252768042846704} a^{3} - \frac{1}{4} a^{2} + \frac{543767816878311957177}{74783546860525993865204} a - \frac{3}{16}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 2411440455.69 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.4.6464000000.7

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.2.2$x^{4} - 61 x^{2} + 7442$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
101Data not computed