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Properties

Label 16.12.2356327674...0553.1
Degree $16$
Signature $[12, 2]$
Discriminant $61^{2}\cdot 97^{15}$
Root discriminant $121.83$
Ramified primes $61, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9590257, -43213225, 38229184, 8333545, -20020144, 4925652, 4265765, -1609224, -398494, 186897, 5874, -11454, 1313, 410, -68, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 68*x^14 + 410*x^13 + 1313*x^12 - 11454*x^11 + 5874*x^10 + 186897*x^9 - 398494*x^8 - 1609224*x^7 + 4265765*x^6 + 4925652*x^5 - 20020144*x^4 + 8333545*x^3 + 38229184*x^2 - 43213225*x + 9590257)
 
gp: K = bnfinit(x^16 - 6*x^15 - 68*x^14 + 410*x^13 + 1313*x^12 - 11454*x^11 + 5874*x^10 + 186897*x^9 - 398494*x^8 - 1609224*x^7 + 4265765*x^6 + 4925652*x^5 - 20020144*x^4 + 8333545*x^3 + 38229184*x^2 - 43213225*x + 9590257, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 68 x^{14} + 410 x^{13} + 1313 x^{12} - 11454 x^{11} + 5874 x^{10} + 186897 x^{9} - 398494 x^{8} - 1609224 x^{7} + 4265765 x^{6} + 4925652 x^{5} - 20020144 x^{4} + 8333545 x^{3} + 38229184 x^{2} - 43213225 x + 9590257 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2356327674777993305467029827040553=61^{2}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.83$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2206} a^{14} + \frac{55}{2206} a^{13} + \frac{271}{2206} a^{12} + \frac{1069}{2206} a^{11} + \frac{329}{2206} a^{10} + \frac{859}{2206} a^{9} - \frac{853}{2206} a^{8} + \frac{501}{2206} a^{7} + \frac{921}{2206} a^{6} + \frac{75}{2206} a^{5} + \frac{245}{2206} a^{4} + \frac{175}{2206} a^{3} - \frac{959}{2206} a^{2} + \frac{877}{2206} a + \frac{35}{2206}$, $\frac{1}{795798719386910948399679044635077360631847288342} a^{15} + \frac{22152848622474811561904776245210581532799882}{397899359693455474199839522317538680315923644171} a^{14} - \frac{78900245459950609489668696816629081578882385527}{795798719386910948399679044635077360631847288342} a^{13} - \frac{76008086832331660664247263281446827872945802151}{795798719386910948399679044635077360631847288342} a^{12} - \frac{95843872007826280473099325651218468906879427531}{397899359693455474199839522317538680315923644171} a^{11} + \frac{13089589412885773968357780425741099627313754431}{795798719386910948399679044635077360631847288342} a^{10} - \frac{102207993387829342135361806576000071708923284995}{795798719386910948399679044635077360631847288342} a^{9} + \frac{98543873780829819509321138738951257293395873934}{397899359693455474199839522317538680315923644171} a^{8} + \frac{60865505687362647932922136108403071970861030767}{795798719386910948399679044635077360631847288342} a^{7} - \frac{153229371359805933438029390388617520561723571039}{795798719386910948399679044635077360631847288342} a^{6} - \frac{198422107123392451668638001429458014826207123984}{397899359693455474199839522317538680315923644171} a^{5} - \frac{90791196298205807739936542601229497018103055653}{795798719386910948399679044635077360631847288342} a^{4} - \frac{15105037990923605165364234071056514071824860615}{795798719386910948399679044635077360631847288342} a^{3} + \frac{12741213281121932384885868497322449433286651165}{397899359693455474199839522317538680315923644171} a^{2} + \frac{234370510767227953122613631994490248677359930973}{795798719386910948399679044635077360631847288342} a - \frac{173039456397479787813125413679561179322329795396}{397899359693455474199839522317538680315923644171}$

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:  $13$
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:  \( 58806365822.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
61Data not computed
97Data not computed