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Properties

Label 16.0.78039659508...8064.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{12}\cdot 17^{6}\cdot 97^{4}$
Root discriminant $98.46$
Ramified primes $2, 3, 17, 97$
Class number $189632$ (GRH)
Class group $[2, 2, 2, 2, 2, 5926]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1171252804, 808621280, 1316168472, 125673152, 466924080, -33396192, 83714568, -7237928, 7714170, -506040, 401124, -15708, 11648, -164, 174, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 174*x^14 - 164*x^13 + 11648*x^12 - 15708*x^11 + 401124*x^10 - 506040*x^9 + 7714170*x^8 - 7237928*x^7 + 83714568*x^6 - 33396192*x^5 + 466924080*x^4 + 125673152*x^3 + 1316168472*x^2 + 808621280*x + 1171252804)
 
gp: K = bnfinit(x^16 + 174*x^14 - 164*x^13 + 11648*x^12 - 15708*x^11 + 401124*x^10 - 506040*x^9 + 7714170*x^8 - 7237928*x^7 + 83714568*x^6 - 33396192*x^5 + 466924080*x^4 + 125673152*x^3 + 1316168472*x^2 + 808621280*x + 1171252804, 1)
 

Normalized defining polynomial

\( x^{16} + 174 x^{14} - 164 x^{13} + 11648 x^{12} - 15708 x^{11} + 401124 x^{10} - 506040 x^{9} + 7714170 x^{8} - 7237928 x^{7} + 83714568 x^{6} - 33396192 x^{5} + 466924080 x^{4} + 125673152 x^{3} + 1316168472 x^{2} + 808621280 x + 1171252804 \)

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:  \(78039659508312536877345552728064=2^{36}\cdot 3^{12}\cdot 17^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.46$
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:  $2, 3, 17, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{70} a^{14} + \frac{1}{7} a^{13} + \frac{8}{35} a^{12} + \frac{3}{35} a^{11} - \frac{3}{14} a^{10} - \frac{11}{70} a^{9} + \frac{2}{35} a^{8} - \frac{1}{35} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{12}{35} a^{3} - \frac{6}{35} a^{2} - \frac{3}{7} a - \frac{11}{35}$, $\frac{1}{33879795841968680729038116752189686722770676417924495553214764270} a^{15} - \frac{30882850160592205171914115468130454216199127533028659346471357}{4839970834566954389862588107455669531824382345417785079030680610} a^{14} + \frac{206051806750226216372394504694106542287470702355636665816594429}{2419985417283477194931294053727834765912191172708892539515340305} a^{13} + \frac{156640760072375084823804991723663917643946167387715212606410398}{2419985417283477194931294053727834765912191172708892539515340305} a^{12} + \frac{2954892561310811445967231273433765967677198155027350555483425311}{33879795841968680729038116752189686722770676417924495553214764270} a^{11} - \frac{573790366057158653330704796633593530444352950960293736371934548}{16939897920984340364519058376094843361385338208962247776607382135} a^{10} - \frac{2594957868618319889031926803122724451328362409521871938528609267}{33879795841968680729038116752189686722770676417924495553214764270} a^{9} - \frac{276554329683986815793746695213340947755757641077071192904601007}{2419985417283477194931294053727834765912191172708892539515340305} a^{8} + \frac{7167032715763961439820741237989613689462127794409423565486383218}{16939897920984340364519058376094843361385338208962247776607382135} a^{7} - \frac{80722278411044623762228333467725625020103437273752542096806296}{2419985417283477194931294053727834765912191172708892539515340305} a^{6} + \frac{484913793627661105752989140367091931896294697513586792075022693}{2419985417283477194931294053727834765912191172708892539515340305} a^{5} - \frac{2779861036372211266629498968153469563334728341785648760282052576}{16939897920984340364519058376094843361385338208962247776607382135} a^{4} - \frac{613682324564409488879237010413868193935525089372501186847097187}{2419985417283477194931294053727834765912191172708892539515340305} a^{3} - \frac{7063121635776310233210578613383078685520580263332726913760091126}{16939897920984340364519058376094843361385338208962247776607382135} a^{2} - \frac{4700647919834817775160455916884898608545785574966505832122926791}{16939897920984340364519058376094843361385338208962247776607382135} a - \frac{364778786466375329664005655712780606636438748987260050646403171}{16939897920984340364519058376094843361385338208962247776607382135}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{5926}$, which has order $189632$ (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:  \( 9072.35800888 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.4.9792.1, 4.4.4352.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$